My neighbor's dog jumped the backyard fence and killed one chicken. He also severely injured the other one. I came out and chased him with a shovel, and he leaped the fence again and ran tearing off into the neighborhood.
I called Animal Control for an investigation immediately. They collected the dead one and said, just so you know, if you pursue this dog could be euthanized."
Oh, I know, because I worked there.
I didn't explicitly have to say much about it to the neighbor. The SPCA was now involved, there are now vet bills that have to be paid so that SPCA does not come after US, and you're responsible.
She covered $667 of the $800 bill for the injured one. I picked up the rest. We both felt that was a fair cost to avoid future conflict with each other, let alone drawn out in public records.
So even if you do not have a legal threat over your dog for the kill, you need to act like you DO for harmonious relations. Because, after all, that chicken could have meant as much or more to that owner as your dog means to you.