Delta wrote a nice little post about who keeps the character sheets in different game groups.
My practice has been for the last three or four campaigns is to encourage players to let me keep their character sheets. I carry my books and games notes in file folder box. I keep the character sheets in a file folder and pull it out when I am setting up. It’s become where players put their notes, maps and handouts I’ve created through the campaign. That has been very convenient when one player can’t make the session and we need some note or another that player was keeping with their character sheet.
I do not require players to keep their character sheets in the folder but I do recommend it. Primarily, I do this because people forget things. I’ve had players forget their character sheet at home, leave it at their friend’s house in the folder with their other character that they are playing in that campaign, grabbed the wrong folder and so forth. I’ve done it myself. If you’ve never tried it, playing a character from memory is a pain in the rear. I can’t imagine it’s even possible if you’re playing a game with a lot of complicated mechanisms and modifiers.
I can see reasons for players to keep their own characters. One obvious one is technology has made it easy to keep track of characters. There is a scanning app that Google makes you can scan an image with your phone in about 30 seconds. If a GM wants a copy of a player’s sheet for reference during preparation, a player can easily scan it with their phone. Most printers come with a flat bed scanner these days so it isn’t too difficult to get access to one if you need it. For games like 5E that have online and tablet applications to manage characters, you don’t even need a paper character sheet as long as you have access to a device that can bring up the character.