I shot a small buck years ago in WV. I was hunting with my 12 gauge and shot the deer perfect through the heart. The buck took off on its death run and out of sight. A minute or two later I hear 2 shots. I follow the massive blood trail right to the guy. The deer was dead in seconds I have no doubt. This guy (and his buddy) claimed he shot it at a full run and ...get this...hit in in the head and chest with his scopped .308. Ok fine...I ask can I look. I lift the deers leg from the oposite side I shot him from and see the massive 1" hole perfectly behind his shoulder. It had passed through and hit his off side knuckle. I wanted to be absolutly sure I had killed the buck so I staid and whatched the guy field dress it. He pulled the heart out in peices. His bullet never found the chest that I could tell. In fact it looked like to me the guy walked up and shot it in the head after it fell.(perfect entry and exit hole in the head with no other damage). The more he talked the more I believed he just wanted to claim he killed one. I doubt he even tagged it realy. He told me, among other things, he had only recently sighted in his gun he had been using for years, I should have aimed higher. He was realy talking alot and I could tell what had happened.
I killed that deer and probably could have made a fuss. But I figured he needed it more than I did. I just let him have it.
So as for hunting on public, or private land for that matter, With a gun I make sure they are down. If I can at all get another round at them I shoot.Especialy if they beed where I can see them.
I think your first shot probably wouldn't have killedhim. I would take that into consideration if Ishot one and someone came up. If I thought the wound to be fatal...I'd give it to them (especialy if it was a younger hunter) but a wound like you describe sounds like it wasn't fatal. He bedded because he was hurt sure. But he probably was just nursing his wound.
You still have your tag so go get a bigger one.
