A long long time ago I was the Town Marshal of a small town Police Department.
Very difficult to prove who owned the "roaming at large" dog(s). No tax tag, no information on collars or suspected Owner denies the dog was theirs. And when the Complainant doesn't want to press charges for fear of reprisal, our hands were tied.
County Humane Society wouldn't respond to "running loose" calls. BUT they would provide live animal traps and pickup captured dogs. So we started catching as many as we could. AND if the Owner went to pick up their dog(s) from the Humane Society they had to pay a collection and feeding fee
So what I'd suggest is to get a live animal trap (buy one or see if the local Humane Society will provide one).
Catch the offending dog. DO NOT try and locate or notify the Owner!
Have the Humane Society come and collect the captured dog. DO NOT tell the Humane Society who you "think" owns the dog(s).
IF your neighbor complains to you about his dog missing, look them straight in the face and say, "Oh! There was a dog that the Humane Society picked up that was after my chickens. I hope it wasn't yours."
The laws very from state to state. Criminally you my not be liable but Civilly you might. If I did shoot a stray dog, I would NOT post it on a public forum. I would not tell anyone. I'd obey the S.S.S. rules (Shoot, Shovel and Shut-up).
Delivering the carcase to the Owner's front step did prove a point but in all probability will start a war. Like you nonexistent Rooster crowing.