I understand about livestock...
... as we have a few horses ourselves (Holsteiners, big ones, and on the expensive side too).
If a dog were molesting the horses, ideally I'd be able to figure out who owned it, and get the owner to deal with it up front. In a more acute case...
I have the luxury of not having any foals at the moment. A dog that gets too stupid with one of our 16 to 18 hand horses will probably get its head kicked off.
But, before that happened, I'd try to catch the dog. Odds are I'd have a gun on my hip, as I normally do, but my first line weapon would be a leash and slip collar; the gun would be a last resort.
I've caught loose dogs ranging from coon-hounds to pit bulls to mastiffs. Most weren't in our paddocks, but you get the idea. If it's feral, you won't get close enough to catch it. If it's a pet, in most cases, it will allow the approach. That's one tip-off.
As noted, I've had dogs get loose before. Not all that often, and despite my measures taken against it. (On one occasion, a neighbor let my dog out of his enclosure to play with her kid, while I was at work...) But my dogs are not deliberately allowed to run loose, nor are most of the dogs beloging to people I know.
In one case, a neighbor had an "underground fence." His dog was a large, unusually aggressive chocolate lab. That dog scared quite a few folks who didn't know about the underground fence, as he'd sprint right up to the property line, but he never posed an actual threat. Another neighbor had a 130lb German Shepherd who was so extremely-well trained that he would not step outside of the yard. "Charlie" used to run around his yard, unleashed and unfenced, all day. Luckily, nobody in our neighborhood back then was afraid of dogs.
I did have one neighbor who kept letting her dog run loose, in a rural neighborhood; it was smallish, maybe 30lbs, but high energy. For some reason, it kept wanting to get into our paddock. I let the neighbor know, but she insisted on letting the dog run loose. She and her husband both acknowledged that a horse might kill the dog, and if so, they'd be out of luck. Happily, that never happened.
So, yes, there are some dog owners out there who make me scratch my head. But there are a lot of cases of dogs that just manage to tunnel under or climb over fences, or slip through doors, etc. Anybody who has owned a pet (or for that matter raised a kid) knows that they can, on occasion, get past the best-laid barriers and plans.