I live out in the country (Ozarks) and my dogs run loose. My neighbors are great people and they know there would be no hard feelings if they shot one of my dogs for chasing livestock. Pets are pets but livestock are a person's livelihood. No problems for 20 years now. Only got 1 call from a neighbor, saying he had to kill one of my dogs but they were all at home at the time. They can, and SHOULD be trained to behave.
Jim