I generally take the same stance as Matt on this issue....
but if its determined somewhow that the animal was still going and someone else put it down then they have the right to the animal. If it was me personally and I put down an obviously mortally wounded animal I would wait to congratulate the guy at the other end.
...in fact, I just got into this discussion the other day with one of the guys I hunt with.
Has it ever happened to me? No. However, it has happened to one of the older gents that I hunt with. He shot a doe during our muzzleloader season and could not initially find it. He was looking in the wrong area. By the time he found the blood trail another hunter was already standing over the deer claiming it was his. After some choice words were exchanged and the game warden was called the two gentleman calmed down and eventually resolved the situation.