I know that in Texas it is illegal to cut the antlers off of road kill, or to take any meat from road kill.
Not sure about a deer that you killed legally.
Not sure about current situation. However, 30 years ago in rural PA, people would listen to the scanner. When they heard of a deer-vehicle collision, they would drive to the scene of the accident to claim the deer for the meat.
Local protocol was that the driver whose vehicle was damaged had first dibs. Then to the first person on scene, etc. Ran EMS for 3 years while I lived there. A couple times nearly witnessed fist fights over a road kill deer...
