You can trap the dog, then call the 'animal control' or sheriffs office to come take care of them. I did this before.Trapping a dog on your property is not illegal they are trespassing and causing damage. And "officer , they may have rabies I didn't want to get near them". I trapped alot of dogs in Ga. that used to be drop offs. The stealing if you are taking to humane society is really far fetched. The owners aren't that savvy, nor do they care about others. I have a few running around that go in my pasture taking trash with them, but not bothering the horses, because I have a donkey and he would go after a dog that gets too close. Most counties have a "leash law" that says they have to be on one or on owners property or control. I also took several 'strays' to the animal shelter and told them I thought they were dump offs. One neighbor had to pay 75$ to bail out (which I didn't know was his at the time, found the puppy crying under my deck on a rainy day), and asked that I not take it again (after me returning it 4 times to no pen yard,and still wanted to let the puppy visit me every day to poop on my porch, dig up my flowers, and cause my dogs to make a rukus. They said "well this is the country" (suburban transplants) I told them how would they like me to turn my horses and dogs lose on their front yard? At my other place I finally fieldfenced the property and am now gradually doing the same, because the trashy locals think they can just throw them garbage and let them run. Or else you get the ones who want little Lassie to be able to "run free" with the wind in his ears and no consequences of his "playfulness". I have about 2 acres fenced with 4 ft fence for daytime and I have a dayyard and aviary with hotwires and roof for night and lockup if we aren't home.
Some Sherriff departments are mis-informed and or not inclined to deal with dog issues. Especially poor counties. Even our poor county in N. Ga. has animal control. I often warn not very bright people there is some guy I heard about who poisons loose dogs (which was actually true)...but the rumor may just help anyway.... Btw aI love dogs, but not irresponsible owners, who force you to be the responsible one, not pleasant , but someone has to be the adult here.....