It sounds like the OP is trying to do the right thing concerning her neighbors dogs. It's a tough spot to be in when you want to be a good neighbor, but have to keep your livestock safe.
I made my feelings very clear within months after moving here that I will not tolerate neighbors dogs coming onto my property, much less chase my animals. A few months after we got here I spotted a neighbors dog (pit bull) chasing some new calves and I didn't waste any time putting a bullet in him. I never told him that I was the one who did it either. He may suspect the dog was killed, but has only said "he disappeared" in passing coversation.
A few months later, maybe a year, we got new neighbors accross the road. A young couple with a couple of small boys. The first time we had a chance to talk after they moved in they were asking about my animals. I went down the list of what we raised and offered to give them the nickle tour. During the conversation I also let it be known that I don't tolerate stray dogs. They understood completely. A few weeks later they got a shepard mix pup that I caught in my pasture and chaseing the cows and horses. I liked this dog cause he was smart and actually did a good job keeping watch on my property as well as his (I never had a deer problem in the garden, or strangers walking up the road for no apparent reason when he was on guard) so I shot at him, not to kill him but to make him think I would. After a couple days he figured out that this only happened when he was after the animals. Like I said, he was smart! He never chased them again. My neighbor knew what I was doing, and they told me that if he was threatening my animals to shoot him. I just couldn't do it to this particular dog though.
Having said that I have another neighbor that can care less. They recently got a hound pup that they'd let run wild every morning. I've caught him twice going after my chickens but as soon as I open the door he takes off. I've gone to speak to them about it and they don't seem to get it. A few days ago I was talking to my neighbor accross the road and mentioned to her that I was keeping an eye out for that dog and if I got the chance I'd shoot him. I think she went over and let the owners know that they better be a little more responsible for their dog or he would be gone. I haven't seen the dog in a few days so I hope they got the message, or got rid of him.
My dogs have never chased my animals, and if they do they won't be around long. Not only for my animals protection, but for my neighbors animals as well. It works both ways.
ETA: I want to also mention that calling the sherriff is a 50/50 proposition out here. They may come out and talk to the owner, or they may not. A stud showed up here one morning and tried to go after my horses. I didn't know where he came from but he forced his way through a fence, injured two of my horses and just raised all kinds of chaos. With my sons help we ran him into a pasture away from the others and I went in to call the sherriff to let them know in case someone reported missing a horse. The dispatcher took all of my information and said someone would come out. They never showed up or even called me back to take the basic information. .