Off. For multiple reasons. IF we are lucky enough to get a truck to it, leave it on until back at camp on the meat pole. But if you are quartering and packing, off immediately. The hide holds in a lot of heat. If you want tasty wildy game, start it cooling down ASAP. Also, leaving the hide on will allow hair to get on the meat. a few wont hurt but a lot of hair will transfer gamy taste to the meat. Skin the hide and get it into a game bag. If you skin/cut the glands or the pelvic area out(which you should) use a different knife or clean it before cutting meat Take care of the meat in all ways from the beginning and you will appreciate it on the dinner table.
I know some folks leave the hide on the quarters in order to keep some of the meat clean. In my opinion, a good game bag is a better idea. And not one of the ones you get for a dollar at walmart. a good cotton one or a sheet is really good. The new material bags are light but i don't think they breath quite as well.
I am always amazed by how many people say "it's how you take care of the meat", and then they don't. I see folks driving back home from a hunt with the hide still on,. Im sure they give packages to their friends who cant understand why you would hunt when the meat tastes so nasty
thats my 2 cents anyway