I highly doubt that you are permitted to shoot a dog on the mere suspicion that the dog is chasing a deer. I suspect you might actually have to see a deer and a dog giving chase.
Otherwise, you're just shooting someone's pet, which will get you charged with a felony and get your right to own firearms permanently suspended, so I suggest that you are careful to make sure that you see a dog actively chasing a deer before you shoot it.
Yes, see 191. Most states that's a police or warden function, and anything you shoot, you're responsible for.
The last few years in WI the DNR has been requesting that hunters shoot feral pigs which are becoming a nuisance. And clearly let hunters know that if you shoot your neighbors pig who happens to have gotten loose, the hunter is responsible.
“Otherwise, you’re just shooting someone’s pet, which will get you charged with a felony and get your right to own firearms permanently suspended, so I suggest that you are careful to make sure that you see a dog actively chasing a deer before you shoot it.”
They are not shooting someone’s pet; they are shooting an animal running loose and endangering wildlife and human beings. If that describes your pet; it isn’t a pet....