I would never shoot someone's family companion because it was simply on my property and I don't care if it was running deer or not. I would find out where it lived and confront the owner. If they care about the dog, they'll take care of it. If they don't, then I guess I'd have to.

My dog is my best friend. He is loyal as the workday is long. I know with no doubt that my family is protected and looked after while I am away. He can find a dead deer no problem, he's also learning to stalk/still hunt with me. Problem is, my dog also has a job. He works at the farm behind my property on occasion when the farmer is having trouble with some livestock. Now I know that I am responsible for my dog, and no matter what. But what if he were to run into the woods to chase a deer when the farmer had let him get bored or something and someone were to shoot him? That would be tragic, but not an accident, and someone would pay, dearly.
I would only shoot the dog as a last resort.