In the Western states where I've hunted (a lot) the animal belongs to whoever shoots the animal and inflicts a serious injury first. I have heard that in the East, where I have never hunted, it belongs to whoever kills it. I can't say that I understand this thinking. If someone else has wounded/crippled an animal and I find it, I'll certainly kill it but I have no desire to claim or tag it.
A few years ago a friend and I leased a piece of property for mule deer. I spotted a buck move into a patch of brush. As I moved down on a stalk the buck tried to sneak out. I went down on one knee, aimed and started to squeeze the trigger, a rifle shot sent the buck out on a stot (jumping run). A guy on the next property, across a canyon was shooting. He shit again and I saw hair fly. I hot on the buck at about a hundred yards and got off a shot. The bullet caught him through a ham and through the guts. The guy across the canyon shot again. My next shot broke the neck. That would be the buck not the idiot about three hundred yards away on another property. He yelled, "Good shot! I'm coming over!) As I made my way to the deer, my buddy moved down from above. He asked if I had done all that shooting. I told him that I had only shot twice, but he could see that I was upset (okay, pissed!). I told him that I had seen hair fly on the guys second shot. I said that if that guy shot the deer first I didn't want it, but I couldn't reward him by letting him have the buck. I knew where I had hit the buck, but I couldn't find the shot that cut hair. Finally, I found the bullet mark. The bullet had cut a line across the back, barely grooving the skin. I squeezed the mark and could barely get a trace of blood. I told my friend that I was taking the buck, and, " when the other guy that shot at it gets over here he and I are coming to an understanding."
When the other hunter showed up, I told him to be clear, it was my buck. He agreed. I asked why he was shooting at a buck on our property. He said, "I saw the buck going out and I didn't think you saw it. I got excited, and the next thing I knew I was shooting at it!"
So this time, no harm, no foul, but if he had hit an inch lower things would have gotten really awkward.