You used to have to change before you left, but all the parks have changed to what they call "cast zooming" so you don't have to change, you can take your costume home (wearing it and all). You are allowed to stop at the grocery store/gas station, etc on the way home from work, but you can't have your name tag on.
CM's get new name tags quite frequently with the changes in the promotions. They are never asked to turn it in, so many CM's still have their old red name tags, or the ones with tinkerbell. They can keep the name tags, not the costumes.