This is a messy problem. The neighbors won't control their dog. The law won't do anything about it. If you do something about it (shoot the dog) you may have both the neighbor and the law on you.
Here's my personal policy on this matter. First build a dog resistant fence around your chickens and other animals. I find that "livestock panels" which are 16 - 20 ft long X 5 ft high panels made of heavy gauge wire, attached to solid posts make a good starting point. I use the ones with 4"squares from top to bottom. These might be called sheep or hog panels. I've then woven long sticks vertically into that, making the height 6-7 ft. Run some barbed wire around the top of that. Then I lay some 3 ft woven fence wire on the ground around the outside perimeter, and hog ring it to the bottom of the vertical fence. This keeps dogs from digging under the fence.
Since I did that, no more coyote or dog problems. Now it's just the occasional hawk swooping down and grabbing a chicken by the head. The head pops off and the hawk flies away with that.
Until you have a good fence your chickens, kids, and other animals will never really be safe. I've had a couple of dogs try to tear thru the fence, but they can't. When I catch one doing that I bust his butt with my slingshot. I'm good with that, but for most people a pellet rifle or paintball gun would be easier.
Shooting to kill shouldn't be necessary if you have a good fence. And it could lead to really big problems with the law and make enemies of your neighbors. If for some reason (dog threatening kids, wife, or my dog) I had to do that, tho, I would just keep quiet about it. SSS = shoot, shovel, and shut up.