How does someone gain ownership over a hunted animal?

First you attach your hunting/harvest tag to the animal.   If you can,  you do not leave the animal and field dress it right then and there.    If you have to leave it a trick I have seen used is putting a personal business card in the throat of the animal.   That way if there is a dispute later and a game warden shows up you can prove you got to the animal first by telling him about your card in the throat.   You can also stick it in the heart if you field dressed it.   Best approach however is to have a hunting companion to back up your claim.

There have been quite a few disputes over a harvested animal. Most often between two hunters that have both shot at the same time at an animal. The hunters will usually look at the entry / exit of the bullet and figure out the angle and who it belongs to. There are times when an animal that has been shot, but doesnt immediately die, that walks on to private property. The landowner may try to claim it as theirs and not allow the hunter access. The hunter then has to contact a state wildlife officer, and show them the blood trail leading to the private property. The DNR officer will then explain to the landowner, that by law the hunter can access the property to claim the deceased animal and usually escorts the hunter on to and off of the property.

Who ever gets there first.

Cos it’s dead 50 yards in front of me

The person who tags the hunted animal first OWNS the dead animal.

In most states you have deer tags on your license that have to be filled out and attached to the animal. When you use all your tags you are done for the season.

You generaly know when you shoot a deer that it is yours. If you shoot it and recover it it is yours.