I think I would call the owner first, if the dog was merely loose and nosing around my place. No dog restraining system is fail proof 100% of the time.
If they were attacking my animals and had already killed one,
I would kill with extreme prejudice. This indicates to me that this dog is not only
able to escape his confines, but now has found a source of amusement when he does and will likely return the next time he slips his barriers.
My dogs are confined by a perimeter fence and an electric system but my older dog has managed to get out a time or two, despite this system. She is completely livestock safe and wouldn't bother another's chickens or sheep. I would appreciate a call from my neighbors if she was merely nosing around their property.
Having said that....
If she were chasing something of theirs (besides the cat) I would expect them to
shoot her straight away and give me a call to come and get her. I would be sad but practical about the situation. She has cost me approx. $1100 over the years in vet bills, even though she was a free dog. I would
still expect the neighbor to cancel her ticket!
I pity the dog that comes on my property, just nosing around or not...my dogs would send them packing with a little less fur and blood to carry along their way!