I'm with the group that says "tell the neighbor" Since we have a trailer park across the street, there is a lot of turn-over. So, when a new dog starts messing with my birds I have the deputies drive across the road and give the owner a warning about the dogs. The second time, no more dog. Then another call to the SO to have them go to the trailer park and give the neighbor the bill for the birds and tell them to pick up the dead dog. If they don't want to come pick up the dog, my "removal fee" is $100.
Sorry, I don't feel generous enough to people like that to let them assume that puppy found a happy home somewhere else. Nope, you are an idiot and because of that your dog is dead.