I live in KY. We have the same rights. Our law reads something to the effect that a dog may run loose during the day, if the dog is not bothering anyone or does not have a history of biting or destroying property. (That includes livestock.) But no dog is allowed to run loose at night, except hunting dogs. The owner has to be out hunting with the dog. But it also states that if the dog is destroying livestock or etc. the landowner has the right to dispose of the dog.
Our county has a leash law too.
You might want to check your state and/or county laws.
I like the idea of the note... But I have tried talking to neighbors in the past about their dog(s). The neighbor(s) have always gotten mad. They always say I can't stand to see the dog chained or fenced up. That is not right to do that to him/her. Well, it is better than seeing your dog injured or worse. If they really love their dog and want to be good neighbors, they will do their part. They may start complaining about your dog or chicken or whatever.
I have never killed anyone's dog but the neighbor(s) think I would. Talking to the neighbor(s) didn't do any good but when they see you walk outside with the gun and take aim. They will call their dog(s).
My neighbor had 5 dogs that started running my horses. I asked them nicely to call off their dogs. They just laughed and let the dogs continue their game. I walked inside and came back out with a rifle, took aim & all of a sudden it wasn't so funny anymore. The neighbor and his family started yelling for the dogs to come back. I never fired the first shot. I haven't had a problem with them since.
I have dogs but they are have their fenced in area. My chickens & etc have their areas too.
I try to be the good neighbor and keep my dogs and etc off the neighbors property. I only ask for them to do the same.
I would check around for some fencing. My aunt bought a dog pen at
WalMart or Rural King. She uses that for her chicken lot. She had coaster wheels welded on the bottom. She moves it from one part of the yard to the other. She locks the wheels and doesn't have to worry about them. It is like a huge chicken tractor.
Good luck with your neighbors!! But if your DH shoots remember the 3S's.