Been through this myself and know your frustration!
We tried talking to our neighbors repeatedly. They're a few years younger than us and kinda just irresponsible. His grandpa (also my neighbor) suggested the "shoot-shovel-and-shutup" method, which really truthfully is the easiest way to handle it, otherwise it becomes a long drawn out thing. My nerves don't favor long drawn out conflict!
They've been nice about it when we talk to them, for the most part. They did put in an electric fence for their dogs, but never trained the dogs to it. Then they started turning the dogs out into a goat-fenced pasture but they still found their way out. Then they tried chaining them, but they keep getting loose (this I don't understand--I had to keep my pitbull on a chain outside for 4 years and she NEVER once got loose).
After a few series of attacks, we resorted to catching the dog and taking it to the pound. We really didn't want to shoot it. Since that, they walk out with their other 3 dogs now and keep them inside the rest of the day mostly. I'm not really confident that they won't be back.
Since the fella shot the one dog like you said, you might try hauling them off to the pound if you can catch them. We took this dog several counties over so he wouldn't come back! If it's a mean dog, I'd either call animal control or if they won't come then I'd shoot it.
Look into your local laws too. Here we have an ordinance that gives you permission to shoot a dog that so much as
harasses your livestock.
All else fails, you might go ahead and fence your property or part of it for the chickens. That's the only way you can be 100% sure. There will always be strays and wild animals to worry about anyway. It seems that chicken losses are inevitable for the most part, unless you keep them shut up in a run that is 100% predator proof.