Rowdy - it's legal everywhere to shoot a dog that is killing livestock. Here, I know where most of the roaming dogs come from - the trailer park across the street. So with a bit of watching, I can narrow it down to a couple homes and I have a deputy come over and inform them that their dog is harassing livestock. Once a complaint has been made, in AR it is legal to shoot the dog the second it enters your property (one of the few states where it IS legal to do that). The longest time a dog was ever kept chained after the deputy visit was 2 days. The shortest time was that the dog was roaming 4 houses down before the deputy even left the driveway. That dog was actually shot that same afternoon by another neighbor. The dog was body-slamming their sliding glass door trying to get to their indoor cats.
Where I was living at the time, it's legal to shoot the dog just for being on your property too, and you don't even have to make a complaint first. I never saw just a visit from the police make someone change their ways either, but the local sheriff's department was really good about fining them for it too, which could be $500-1,000 in a very economically depressed area. That usually made them shape up for awhile at least. It wasn't illegal just to let your dog roam, but it was illegal to let them do it if they were a nuisance and harassing wildlife or livestock.
I was friends with one of the sheriff's deputies and apparently I was actually known as a bit of a PITA for harassing the cops to do something rather than just shooting the dog. I was like, hey, I pay taxes, I don't want to have to shoot dogs! It usually wasn't too hard to figure out who the dog belonged to, but that one dog we just couldn't figure out. I did eventually learn who owned him purely by accident months later, even, but of course by then it was too late to do much.
I think that people who haven't seen just how vicious and destructive domestic dogs can be when loose and unsupervised just don't get it. And it is sad, because they're just being dogs and it's their owners' fault that they're in that position. I don't want to derail further though, I just brought it up because I wanted to illustrate that even in areas where there isn't heavy traffic, it's stupid and dangerous to let your dog roam.