Like it or not, this is how it goes in farm/rural (NOT suburb) country. Dogs do run loose. BUT, anyone who lets their dog run also knows that if said dog is caught/seen killing farm livestock, said dog WILL be "terminated with extreme prejudice", and more often than not, the dog owner will agree with the action. The criterion isn't "loose dog", but "killer dog". Dogs that leave livestock alone are themselves left alone. No harm, no foul.
And yes, farm folks do lose pets to traffic (a much more frequent occurrence than the above "termination").
In addition when I lived in NYS it was not uncommon to see packs of dogs chasing down deer. It was perfectly legal to shoot the dogs.