I saw an episode of Dirty Jobs on the Discovery Channel once where Mike Rowe worked for a day with the guys that have that job. They work for the state DOT I believe, at least in the state where he was. Some of them may work for the county. Others may be privately contracted by the local government.
In America the deer are suppose to be picked up by whatever town it was hit. Sometimes they take there time in doing so. Also, whoever hit them, if they have a deer license they can take the carcass themselves.