I have never been able to catch a dog that was running deer on my properties.
I had a neighbor last year that let their dogs breed like rabbits, and run wild. They were up to like 13 dogs in a pack running a square mile like they owned the place. That was a pretty serious problem.
If you know the owners, just lie. Tell them you run yote traps, and if their dogs get into them you may not check them in time before their dog dies a slow and painful death. If they care about their dogs they have to keep them off your private property. If you can catch them and they're nice you can take them to the owner and ask them to remedy it. If they don't next time you just go drive them off and dump them at a pound.
I do suspect if a giant buck got ran off in front of me by 3 dogs. There would be lead flying though.
My father bird hunts, and goes through a lot of trouble to keep his dogs put up. They got out before, and all of them died of poisoning. That was heart breaking for him. So I would always try to work it out because some people do actually care, and I would be much more understanding of dogs digging out once or twice for an old man trying his best vs someone that doesn't care and won't work with you.
Of the 4 properties I hunt. I had dog issues on all of them last year, and I haven't seen a single dog this year. I'm not sure where they all went.