I encourage players to give me copies of their sheets, but I don't require it. If I did, I'd be forever wanting more and more copies as levels were leveled, and perpetually forgetful people would forget them.
I look over sheets now and again for any glaring errors, but I don't go through it with a fine-toothed comb. I would, though, if I suspected chicanery or other such foolishness. If I FOUND badness that could not be a simple mistake, there would be much frowning and head shaking from me, and a hefty XP penalty for them.
I wouldn't like it if a DM insisted on keeping my character sheet and making all the adjustments to it. I like fiddling with my own numbers, and I go through it every once in awhile to make sure that I don't have any mistakes. If they want to check it over, cool. If they want a copy in case I'm absent or to plot some doom, cool. But as long as my information is correct, it's my sheet, durnit, and Ima keepin it.
One thing I DO insist on is that my players have their character sheets on CHARACTER SHEETS. a couple of my players have a habit of scribbling their stats etc onto looseleaf in barely legible chicken-scratch, and I can't cope with that. If I do need to look their sheet over for something, I want to be able to see what I'm looking at.