For instance, how did they go about implementing the Statute of Secrecy? Imagine if they'd tried to not only "disappear" all witches and wizards and magical beasts and beings from public view (at least, as magical people), but also Obliviate every Muggle who'd ever seen magic? That would be a logistical nightmare. Because this isn't your average Obliviation, where you just vanish the last few minutes or hours of memories - you'd have to sort through their whole lives and delete specific memories, or just wipe most of their memories entirely and give them new ones, either of which sound much harder to do. Probably very few could actually manage that complex of a memory spell. And even if they were mostly trying to be discreet about magic beforehand due to the witch hunts, there could still be a lot of Muggles to Obliviate.
I've stitched together three essays (the first and third from several years ago, the second from yesterday) trying to figure out a progression of events and how this all actually worked.
So, first up, what finally convinced them that secrecy, not just discretion, was necessary.
The Inciting Events
One of Dumbledore's notes in The Tales of Beedle the Bard mentions that (contrary to Harry's textbook's assertions in PoA) while many witches and wizards escaped the witch-hunters, they weren't so superior to Muggles that it was only Wendolyn the Weird-style fun and games. Nearly Headless Nick had his wand taken from him, which is why he couldn't prevent his neck getting messily chopped. One assumes he's one of those wizards who never got the hang of Apparition. And - perhaps more significantly - many of those finally caught and executed were children, who couldn't control their magic yet. This disaster led to the establishment of the International Statute of Secrecy.
No wonder Harry's textbook takes such pains to convince the kiddies that Muggles are harmless. "Actually, they nearly wiped out wizardkind altogether in a genocidal attack on children like you, and we don't dare let them know about us or it could happen again" is hardly likely to inspire a desire for peaceful co-existence. Inspire them to join Grindlewald or the Death Eaters (depending on the era) to kill and rule over Muggles, more like. No wonder all kids are confined to Hogwarts, Middle of Nowhere, for most of the year instead of getting frequent field trips. (Surely the Muggle Studies class, at least, would benefit from visiting Muggle areas to practice blending in, but Hermione mentions no such thing.)
This also got me thinking: we know that Harry's trial before the Wizengamot was unusual, but maybe it was just unusual for the times. After a few generations had gone through Hogwarts being taught that Muggles weren't dangerous, they probably didn't take breaches of the Statute of Secrecy as seriously as their great-great-grandparents. But given how serious such breaches could be back in the day, maybe Fudge was just enforcing the full consequences as established back in 1689/1692 (not generally used in modern times, but still on the books). Even in ordinary circumstances, it is apparently normal procedure for a minor doing magic around Muggles to get one warning, then a snapped wand and expulsion, which is essentially being stripped of most of one's rights, social standing, and possibilities for employment. That's still a pretty tough punishment.
Also, the Decree for the Reasonable Restriction of Underage Sorcery wasn't written until 1875. Now, on the surface it sounds like this decree further restricted minors' rights to do magic, but I wonder - could it have been the new and improved version of an older, harsher decree? Maybe 18th-century kids wouldn't have gotten a warning, but would have been expelled on the first offense. And so they eventually decided to be more reasonable about their restrictions. The one allowable exception we know of to the no-magic-before-Muggles rule is for self-defense - just what you'd expect if your goal was to keep the kids from bringing persecution on the magical community without taking away their ability to protect themselves.
This also might make sense of another Potterverse detail, if I remember my witch-craze history right. I think that it was more intense and long-lived on the Continent, and that the later phases focused on demonic/witch children at least in some places. This backs up Dumbledore's note, and the uneven intensity of the witch-hunts would explain why while Hogwarts is in a remote location and has Muggle-repelling charms, Durmstrang is so much more secretive that Karkaroff doesn't even want Krum talking about the school's general geographic surroundings to other wizards. How much can "trees and glaciers" narrow it down? Especially since Durmstrang seems to include both Germanic and Slavic populations? But perhaps historically, their experience was so much worse that they decided one can't be too careful.
I'm also now curious about exactly how many magical children did get caught and executed. We know the wizarding population isn't huge; it could have been a demographic catastrophe. Maybe the wizarding population was actually much higher, in proportion to the Muggle population if not in raw numbers, before this era.
Next, the life of a famous historical figure might give some clues to how they went about implementing the statute.
Sir Isaac Newton, Alchemist
I was reading this article - Isaac Newton, World's Most Famous Alchemist - when an odd detail caught my eye. It says, "Principe notes that Newton suffered a mental breakdown a year after Boyle’s death [in 1691] and wonders if that episode might have been brought on by mercury poisoning... But Newman thinks that Newton’s breakdown is just as likely to be related to Locke’s trying to set him up with a well-to-do widow."
So. Newton's alchemy partner dies and he has a nervous breakdown. Sounds understandable enough, if he thought now he'd never manage to make a philosopher's stone and join the ranks of Nicholas Flamel and... well, Nicholas Flamel. But the timing leaves some interesting possibilities for the Potterverse!
Consider: the Statute of Secrecy was apparently drafted or maybe signed in 1689 and put into effect in 1692 - the year Newton had his breakdown. I would imagine that if he had close ties to both Muggles and wizards, the statute could be a major stressor. (Can you imagine having to Obliviate all those great conversations with the Muggle friends you had trusted to be discreet? Awful.) Plus, if Locke was pressuring him to marry a witch he didn't like simply because marrying a Muggle was not looked on favorably at the moment and he had a duty to propagate magical traits or whatever, that could be stressful even if Newton had been inclined to marry as a general principle. (Since iirc Cambridge required fellows to be ordained - which Newton managed to wriggle out of - and since I've heard fellows were unmarried before the 19th century, maybe his reluctance to marry had as much to do with liking his job as not liking women.) So, not only has he had a major setback to his chances at making a stone, he also has to live a double life and dodge his friend's attempts to marry him off to that awful Cornelia Black! It's no wonder he cracked a bit.
(He was discreet about his alchemical work even before this date, but that might have more to do with the King not being too keen on anyone learning how to make gold and thus screw with the economy, which is why some alchemical practices were technically illegal.)
Newton obviously remained prominent in Muggle society, so either he cut ties with wizards somehow, or he was allowed to keep his identity so long as he didn't do any actual magic where Muggles could see. Which actually makes sense from a strategic standpoint: what better way to get Muggles on the track of deciding magic isn't real than to have an actual wizard there distracting them with math and science? And what do you know, in 1693 he started publishing his work on calculus, which shortly led to the HEY GUYZ I TOTALLY DISCOVERED CALCULUS BEFORE LEIBNIZ controversy. And then he went on to run the Muggle mint. No goblins here, no, of course there's no such thing, your majesty! What ever gave you that idea? And I can assure you, those counterfeiters I caught absolutely were not using any supernatural methods of reproducing coinage, no way. And psst, wizard friends, you might want to explain again to Julius here about the not letting Muggles get ahold of magicked objects part again... Finally, some of his papers were destroyed in a laboratory fire, and others (the Portsmouth Papers) disappeared from view until the 1930s, when they could safely be passed off with, "Isn't it interesting how silly superstitions like alchemy paved the way for modern chemistry!"
So, they reduce their logistical nightmare to manageable proportions by only disappearing those witches and wizards they can at the moment, Obliviating the memories they can get ahold of, and leaving some magical people in the Muggle world to slowly get Muggles to believe that whatever their grandparents say they saw, it couldn't have been real.
And it seems to have made a difference within a few decades, based on what happened to the "godless children" of Augsburg.
The Child-Witches of Augsburg
This incident comes from Lyndal Roper's Witch Craze. To summarize: in the early decades of the 18th century in Augsburg and other German areas, the general fears turned from old women as witches to children as witches. Parents reported that their children sprinkled shards of glass and devilish powders in their beds to sicken them, held witch's sabbaths, and played all manner of Satanic games (among other things). Many of these children ended up incarcerated in a hospital for years.
Now, probably many cases were just disturbed children from dysfunctional families. However, in the Potterverse, the remainder of the cases may have been something more. The wizarding world had retreated into secrecy and seclusion around 50 years ago. By the early 1700s, any magical children born in the Muggle world most likely would be cut off from any magical training. However, wizards had lived openly recently enough that stories about them would still be circulating. Some children - magical or not - could well try to imitate the spells they had heard about from their older cousins or aunts or whomever. Some of them might even have succeeded. Regardless, their parents would have worried.
Fortunately, the practice of executing witches had pretty much died down by now (with a very few exceptions). Even though at least some of the children confessed to attending witches' sabbats - which would be plenty to have gotten you burned before - their hospital jailers went, "Well, we were watching and it looked like they were in the room the whole time they said they flew off to join that Black Mass, so maybe they're just crazy or making it up." The state eventually sent the kids home and told their parents to deal with it, and people started agreeing that the whole witch craze was perhaps a bit over the top. Belief in witches died down as (accurate) memory of openly practicing witches and wizards faded, and by the end of the century witches were little more than a fable to many Muggles. (Not all, but it was excellent progress.) Any magical children born in the Muggle world after that would only manifest magic spontaneously, not deliberately try to mimic traditions. Hogwarts (and perhaps Beaubatons) tried to locate and assimilate as many Muggleborn magical children as it could to keep even too much spontaneous magic away of Muggle eyes; whether or not they were on top of this from 1692 or whether they refined their techniques we don't know. Durmstrang seems to have decided that they don't want to risk Muggleborn children attending, with all those leaks about magic to their Muggle parents and who knows who else. (And some of them are pureblood fanatics for other reasons, of course.) We also don't know whether they decided the risk of Muggleborn magical children having spontaneous outbursts is simply less risky than trying to bring them into the magical world (and since Durmstrang apparently has students from Germany, Scandinavia, and Eastern Europe, the concentration of magical births might be fairly low), or whether another, less prestigious and newer school in the same region takes the Muggleborns.
And so even with imperfect secrecy, within a few decades the wizarding world managed to convince the Muggle world that maybe they were being too credulous about this whole magic thing, and maybe they should have an Enlightenment and play with math and science instead.