I'm really glad you posted this thread because this exact thing happened to me last week. Leaving work, headed down middlebrook pike my co-worker hit a dog. I stopped to help her because she was very upset.
When I got to the dog he was still breathing but obviously very very bad off. I called 911 to ask if I was allowed to put the dog out of its misery. Before I got an answer I was disconnected.
Just a minute or so later the dog passed away, I pulled him to the side of the rode and called the phone number on the tag, the owners dad showed up and I helped him load it into his car.
I would say that it it's the city limits it's probably best not to discharge a gun, I'm glad I didn't have to.
Now the creepy part, found out the next day that the dog was a friend of mines. Talk about a small world. She was really glad I called.