the only way to 100% be able to keep peace with a neighbor who lets their dog roam is to let the dog kill your chickens anytime it wants. There are people who will take offense at the very suggestion that their precious widdle puppy would do such a thing.
That's why I let the cops handle it. I call and make a complaint, ask for a report of the damages. If I know what house the dog came from, the officer goes over and lets them know that it is legal to shoot the dog and advise them to keep it contained. We have several farms/people with livestock in the area so there is no way to know who called (except when they live across the street and see the cops). On the other hand, people tend to take it more seriously when there is a police car in their driveway. If I see the dog again, then I have done what I can to inform the owners. I don't need to worry about photos or evidence or anything like that.