I have done some editing.
I have read that anybody can edit Wikipedia but thought nothing of it until now.
Not completely true in the way it sounds. Overwhelmingly most people can edit overwhelmingly most pages. There are pages the editorial staff considers subject to "vandalism" that are locked and can only be edited by allowed people.
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There is a core editorial staff and they monitor the editing, but not someone goes out and fact checks every edit every person puts in. That is left to the people who regularly visit or watch (get automatic notices of updates) for a page. By and large, the editors exert control over this process rather that directly over content.
And through this process, people get banned. Of course, the Internet is inherently anonymous, so it is pretty easy to try to sneak back in. Then again, that is a well known fact, so savvy editors can block several ways of sneaking back in. Especially the ones used by novices.
So I guess I am qualified to edit a page on a king I knew pretty much nothing about.
The presumption about allowing editing is not that you are qualified, but that you adhere to Wheaton's Law. If that is demonstrated not to be so, the situation is handled as mentioned above.
How often does this REALLY happen?
If you mean how often does some random person post some tripe they should have, the raw numbers are not tracked, but I would guess about 100.05% of the times it is corrected within a few minutes.
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So, here's the upshot. Next time that somebody complains loudly that anybody can edit any page in Wikipedia and put anything they want in it offer to bet them $100.00 USD that they are not right. For this type of person, it is best to get their assertion in writing. Then go to Wikipedia and search POTUS. Say, "Go ahead. Edit." The President of the United States is one of those pages which is protected, as are a lot of other sitting political persons and recent past ones.