Whitetailed deer are a native species, so if it escapes someone's ownership and get killed during hunting season...finders keepers.
The difference is with non-native species. If an elk gets loose and gets killed by a hunter, there is no season or regulations for them in Ohio. That means you could shoot it with a weapon of your choice, day or night. The only recourse the owner would have it to file for personal property damage, to recover the cost of the elk. However, that very seldom happens.
Bowhunter57