I also am flabbergasted by the folks who think it's great to have your dogs to kill, they will do so indiscriminately. We are the humans, so let's try to at least appear more intelligent when making decisions.
If my own dog killed my livestock, I would shoot it myself unless I could find an immediate home for it with someone responsible who lived in town. If my dog were running deer, I would put it down, too. It is
illegal to let your dog run deer or harass livestock belonging to other people, even if the livestock on on your land! (Open range here).
That said, my dogs are intelligent enough to tell the difference between mice, packrats and pocket gophers and poultry, dairy goats, etc.
Very few dog owners are willing to adequately compensate the owner of killed livestock or poultry (if at all). Also, stock doesn't have to be killed to affect it's health, performance or production. Dairy animals stress and drop in milk production. Some animals will spend so much time in fear of the dogs that they don't get to graze or eat enough. Pregnant cattle can miscarry due to being chased by a little dachsund.
And then, most of the dog owners will tell you that Rolfie didn't "mean it". He just "likes to play with/visit the animals". And they might be sorry, but are they going to make it right or at least keep their dumb, marauding canine at home? Probably not.
I just changed my mind. I don't feel bad about shooting dogs. If my dog comes over and bothers your stock, I would understand 100% if you shot him or her. Would I be sad? Sure. Would I blame you? No. Would I be ashamed of myself as a pet owner and neighbor? YES, I would be.