I can sympathize with your dilehma. When dogs get used as pawns in personal attacks it can get ugly in a hurry.
Nobody wants to shoot a pet. And I do try to communicate before acting. But I have no qualms about it. And I have only shot dogs for one of 3 reasons. One, to put it down to be humane. Two, If it habitually(more than once) kills other pets or livestock(chickens,rabbits,ect.) and Three, if it is maliciously vicious towards human life.
What I do have issues with, is people who get dogs and keep them tied up or caged. I for one will not own a dog if I had to do that. Now you do run a risk of someone being less than pleased with your free running pet, and they can get themselves into trouble, but a dog worth anything will be trainable and can be taught boundaries and restraint.
In the past I have had a dog picked up by animal control, Lost one on the highway, and had another came home gut-shot. In hiensight, The one that got picked up was a doofus and had a reputation for doing most anything for tablescraps, The one that got ran over on the highway was 16 years old and was about 80% deaf and blind in one eye so he probably didnt see it coming(literally), and the one that came home with a deer slug in his gut was a deer runner (a bad trait I couldnt get him to shake) and I am sure he rousted some deer under the scope of a hunter in a stand. In wich I would have shot him myself. Anyway, considering those odds, I have had very little trouble. And most dogs are harmless and dont merit the extreme of being eliminated. If you train your pets correctly and express to them love and consideration(and lots of milk bones), they are content to stay on property and hunt at night. Of course, I live in a rural area and my neighbors are limited so it is less likely to ruffle feathers than if I were to live in town.