In Michigan, I think legally only a game warden or police can dispatch a deer struck by a vehihcle. I believe the driver may claim it if law enforcement has been notified. I had a cop offer me a road killed deer last winter just outside of town, but still in a heavily populated area. It was late on a Sunday night and I didn't have time to deal with it so I passed. He was getting off work in a half hour and was going to go back and get the back strap if he couldn't find anyone to take it. Generally now the authorities just let them lie on the side of the road instead of hauling them away like they use to. At one time it was common for the RK deer to be given to the needy but I don't think that is done much anymore.
Having said that I know people that have put an injured deer down discreetly and just keep quiet. Some have harvested what was salvagable. A college classmate hit one one night and called the game warden. The deer had broken legs and was strugglig but still very much alive. It was during hunting season and the GW was busy quite some distance away and said he didn't care what the guy did with the deer. My classmate looked in his truck and all he could find to dispatch the deer was a base ball bat to dispatch the deer with. He said it got a little messy.
Although law enforcement often turns a blind eye towards citizens dispatching injured dee,r some people have gotten in trouble. A local farmer had a road killed deer at the end of his drive and called the authorites. They said to let it lie. As it was starting to smell he took his backhoe and dug a hole next to the road and burried it. I forgot what charges they brough against him, but he was chagred, as the dead deer was property of the state.
An amusing story from a few years ago came from Michigan's upper pinnsula. Many places in MI UP are extremely poor. It was a bad winter with lots of snow and in some areas the deer were herding up in the roads. These area became hot spots for accidents and poachers. The GW was checking the area out when he noticed a car driving irratically and swerving around, turning around, and repeating the process. The GW saw the car hit a deer and continue driving irratically. He pulled the car over and talked to the woman driving the car that was full of dogs. She was arrested as she had intentionally hit three deer. She was harvesting dog food.