Fieldmouse,
Ohio requires you to tag the deer where if fell, transport it to a deer checking station (there's several in each county), the check station removes your tag and replaces it with a DNR metal tag.

After that you may do with it as you please, unless you're having it mounted, then the taxidermist will need to see the metal tag. Also, at a meat processor, they want to see the paper work for the deer....as to whether it's a road kill (issued by any law enforcement officer) or has the metal DNR tag that shows it's a hunting harvest.

Correct me if I'm wrong, but I believe in Michigan you have the option of checking the deer in or not. I've heard of guys taking deer to be processed or to a taxidermist that were taken in Michigan and had to show the deer tag that they had purchased, ONLY. This is a "loop hole", in my opinion, that would allow you to shoot a deer in any state and get it processed or mounted in Ohio....not that that's a bad thing.

Good hunting, Bowhunter57