People have a right to defend themselves, their families, and their livestock from animals. But unless a dog who wanders onto a neighbor's property was posing an immediate threat, there is probably no legal justification for shooting them. Unfortunately, that doesn't stop some people from pulling out a weapon and killing a neighborhood dog (or cat or other pet).
It is true that some areas might have leash laws that forbid dogs from wandering. That still isn't likely to give someone the right to shoot the dog, except in self-defense or to prevent the dog from harming someone else (depending on the precise terms of the statute, but any other approach is basically unheard of).
You do have the right to sue a neighbor who wrongfully shoots your dog, most likely in small claims court. The advantage of that forum is that procedures are simple and you don't need to hire a lawyer (the costs of which would quickly exceed any amount you're likely to win).
Of course, suing probably won't do much to relieve your anger and sorrow. It will certainly not make you wealthy. Some judges will take into account the sentimental value of a lost pet in awarding monetary damages, but others won't. The most you are likely to win is the dog's "market value." That is very little for most pets who aren't young prized breeds, or perhaps Internet sensations in their own right.
And if you were partly at fault (say, by leaving your dog off leash, or not controlling its barking), the judge might decrease the already small amount you can get from the neighbor.
For more see Nolo's Small Claims Court Resource Center.