ani_bester had a question regarding why Wizards even bothered telling Muggles of their existence. I ended up having a bit of a post-fest with maelwaedd, which resulted in this small essay.
Who in the Muggle government knows about their existence?
Obviously, it's got to be kept to a certain few. I'd wager that it varies from nation to nation, depending on their technological advancement, the amount of land they govern, and even their own ties to magic in their general culture.
One thing that was brought up was what happens to Muggleborns or mixed bloods who want to melt into the Muggle world? Another one that I've pondered at work was, would such a system have a plan to cover this?
Let's say, for instance, that Hermione's parents tell their co-workers that Hermione goes to a prestigious school in Scotland. Is it still called Hogwarts, or does it have another name? If there's some sort of system wherein the school's names are kept (and I really can't imagine there isn't one), how easily can you look it up?
But more importantly, who organises all this?
Obviously, the brunt of the load falls on the Muggle government, since the Wizarding World is aware of the Muggle World, but generally, it doesn't go the same for them. What would happen if, say, Mrs Wigglesworth became irrationally worried that Hermione had, in fact, been kidnapped or something, and she went to go check up on Hogwarts? Would she be able to find a false government file that states 'Hogwarts is an ultra-exclusive school in the North...' While Hermione's parents can get away with saying their daughter earned a scholarship, not every Muggleborn parent is going to have a child so sharp that that idea is easily digested.
So I wonder: is there a section for most of the major offices that has a liaison? I can imagine some poor, harried soul working for the Education department who's frantically trying to create a system equivalent to OWLS for people who want 'Muggle' report cards to show a uni, if they decided to go a Muggle uni. Does the FBI know about this? Does the NSA has a special wizarding department that takes care of small disasters? Perhaps the Wizengamot and MI-5 have semi-annual meetings? Who could possibly organise all this, and how? Or perhaps there's a separate branch that deals exclusively with Wizard-Muggle relations.
And then, there's the idea that some people are going to want to melt back into the Muggle world. Do they get school certification equivalent to a high-school diploma or GCSE, TGAU or Standard Grade? Do they have someone who deals with that? Do they have a program to re-introduce you into Muggle-life and things you may have missed?