I was raised that a deer belonged to the person who drew first blood too. But these days, it's pretty much whoever shot the deer last who owns it.

Like, if you shot a deer, and it ran around for a week, and someone else shoots it and puts it down, it belongs to them. That's kind of extreme, but if you shoot one and it runs 100 yards to someone else and they get another bullet in them and knocks it over dead... It's theirs. Unless you can somehow know which bullet hole was yours and which one was more fatal.

Back home we always just left it up to a coin toss.