A lot of it depends on the state. Some of the southern states (where the woods are very thick) - allow using dogs to hunt deer - but most of the northern states do not.
I do know, from personal experience, that when you let dogs just run loose in the woods they will get together with other dogs and form packs and they get into all sorts of mischief - I have seen a pack of 5 or 6 dogs - (now these are peoples pets that are running wild - Collies - Labradors - Beagles - Spaniels - and larger hounds of all kinds) run into my cousins barnyard and chase his cows and sheep all over the place. They had killed and crippled several of his sheep just a few days before I saw them, and he and I went out and shot 4 of them before the rest got away.
In northern Michigan - when I was a kid, I worked in the woods cutting pulpwood for a while, and we always had our rifles handy to shoot dogs running deer. This was back in the late 1950's and it was legal to shoot them back then, but it is not legal to do it there now. Now, you have to report the dogs to the Game Warden and he takes care of it.