A guy on EN World made the distinction that there is a line between "stuff PC should know" and "stuff PC wouldn't know". So eg a PC talking about his family and home village would be on the PC side of the line and would be legitimate for the player to add in play - the GM could negate it by saying later "You thought that, but actually..." I remember one PC Hakeem, it turned out his dad probably wasn't who he thought it was, rather another guy who was now an evil warlord.
Does anyone object to a player adding that sort of stuff? I do think it can be taken too far (for a traditional game); one (otherwise v good) player came up with a lot of convoluted family stuff with relationships (ok) but also with plots that didn't interest me to play through and didn't relate to the other PCs. I rem saying "OK, that stuff gets sorted out". If she had just made NPCs that would have been fine.
I don't object to a player adding "color" to a character. That the character disliked a childhood acquaintance is color. What the character did about it is probably mere background. That's no different than a mannerism the character affects. If there is any meaning to attach to it (e.g. what the childhood acquaintance did and certainly why), then it falls under the GM umbrella for me.
Again, in a one-shot or in series of unrelated adventures that might as well be one-shots except with the same characters, I don't mind more than that. When I'm running my usual campaign, details about who you know, what they did, why they did it, etc. often form a web of intrigue that if pursued can drive a major course of action in the campaign. For example, in a recent D&D campaign, one of the threads was that three of the characters shared the same mother and none of them knew it when the game started. That was why all three of them had some unusual abilities.
Of course, you need players that thrive in that kind of environment. The one thing that unites all the players in my main group is that they love uncovering mysteries. The last thing they want to do is provide that information on the fly.
Edit: It's a good thing that the players agree with me, because I don't particularly enjoy the whole, "Player throws a monkey wrench into the basis of the campaign and now I need to somehow work it in," activity. Which might be strange, because I don't mind monkey wrenches tossed by actions of the characters.