Here's my 2 cents worth...
I've had to deal with goat/chicken/rabbit etc. killing dogs for many years. What I've learned is that having a good high, tight, strong fence is the best protection. On another post I talked about weaving 6 ft long salt cedar or willow sticks (every 4" or so) vertically into an otherwise flimsy wire fence to make it dog & coyote resistant. Then lay some scrounged fence wire on the ground outside your fence, attach it with hog rings or wire to your existing fence. Then shovel some dirt over it to hide it. Now dogs can't dig under your fence.
When it comes to shooting dogs there are some serious issues to consider.
First is safety. If you shoot a gun at something the bullet, pellet, bb, shotgun pellets, paintball, etc could keep right on going and hit somebody or something you don't want to hit. Don't shoot at a dog that has anything you don't want to hit behind or beyond it. Also keeping a gun around the house is a huge responsibility, especially if you have kids. Think about this several times before you get a gun. How will you keep it stored so kids can't get to it? Are you within city limits, where it may be illegal to shoot a gun except for self defence?
Second is how shooting dogs will affect your relationship with your neighbors. They may take revenge on you if they know you shot their dog. You may be justified, but it could still cause trouble.
But it sounds like you already have trouble. Dogs killing your animals and neighbors not willing to take responsibility for the dogs puts you in a position where you either have to give up and leave, give up on having animals, or defend yourself and yours. My choice is option 3. What I would do is buy two guns and get serious about learning to use them safely and effectively. The first would be a .22 rifle (I recommend a Ruger 10/22). With that you can defend yourself and your animals effectively. It's accurate and will certainly kill dogs (tho not quickly) and you can use if for home and self defence if necessary. It has no recoil, isn't loud, and ammo is cheap so you can learn to use it without going broke. But I would only use it as a last resort. The second gun would be a decent pellet rifle (and that may be what you already have). With that you can sting dogs' butts out to 50 yards, and do it quietly. Once a dog has had his butt stung he will probably avoid your place in the future. It's true that the pellets may penetrate the skin and cause infection. Then the owner will have to take him to the vet and spend a bunch of money having the pellet removed, antibiotics, etc. After that the owner will probably see that keeping his dogs home is in his best interests. An alternative would be a paintball gun, which won't do any serious damage, and hurts like heck. But they aren't very accurate beyond 20 yards or so.
Once you've taken care of your fence and if you decide to arm yourself and start defending yourself and your animals, I would just shut up about it. Don't draw attention to yourself by calling the cops, complaining to the neighbors, etc. Just pop any tresspassing dogs in the butt with the pellet rifle, and shoot any killers with the .22. The place to shoot them is right behind the shoulder, thru the ribs, into the lungs and heart. It's a bad, and sad, thing to have to do, and I hope you can avoid doing it.
I've had to deal with goat/chicken/rabbit etc. killing dogs for many years. What I've learned is that having a good high, tight, strong fence is the best protection. On another post I talked about weaving 6 ft long salt cedar or willow sticks (every 4" or so) vertically into an otherwise flimsy wire fence to make it dog & coyote resistant. Then lay some scrounged fence wire on the ground outside your fence, attach it with hog rings or wire to your existing fence. Then shovel some dirt over it to hide it. Now dogs can't dig under your fence.
When it comes to shooting dogs there are some serious issues to consider.
First is safety. If you shoot a gun at something the bullet, pellet, bb, shotgun pellets, paintball, etc could keep right on going and hit somebody or something you don't want to hit. Don't shoot at a dog that has anything you don't want to hit behind or beyond it. Also keeping a gun around the house is a huge responsibility, especially if you have kids. Think about this several times before you get a gun. How will you keep it stored so kids can't get to it? Are you within city limits, where it may be illegal to shoot a gun except for self defence?
Second is how shooting dogs will affect your relationship with your neighbors. They may take revenge on you if they know you shot their dog. You may be justified, but it could still cause trouble.
But it sounds like you already have trouble. Dogs killing your animals and neighbors not willing to take responsibility for the dogs puts you in a position where you either have to give up and leave, give up on having animals, or defend yourself and yours. My choice is option 3. What I would do is buy two guns and get serious about learning to use them safely and effectively. The first would be a .22 rifle (I recommend a Ruger 10/22). With that you can defend yourself and your animals effectively. It's accurate and will certainly kill dogs (tho not quickly) and you can use if for home and self defence if necessary. It has no recoil, isn't loud, and ammo is cheap so you can learn to use it without going broke. But I would only use it as a last resort. The second gun would be a decent pellet rifle (and that may be what you already have). With that you can sting dogs' butts out to 50 yards, and do it quietly. Once a dog has had his butt stung he will probably avoid your place in the future. It's true that the pellets may penetrate the skin and cause infection. Then the owner will have to take him to the vet and spend a bunch of money having the pellet removed, antibiotics, etc. After that the owner will probably see that keeping his dogs home is in his best interests. An alternative would be a paintball gun, which won't do any serious damage, and hurts like heck. But they aren't very accurate beyond 20 yards or so.
Once you've taken care of your fence and if you decide to arm yourself and start defending yourself and your animals, I would just shut up about it. Don't draw attention to yourself by calling the cops, complaining to the neighbors, etc. Just pop any tresspassing dogs in the butt with the pellet rifle, and shoot any killers with the .22. The place to shoot them is right behind the shoulder, thru the ribs, into the lungs and heart. It's a bad, and sad, thing to have to do, and I hope you can avoid doing it.