What to Do About the Neighbors Dog?

My advice is definitely talk to your neighbors. If you simply rehome their animal (regardless of the apparent treatment), they could come back and sue you. I would explain the situation, and tell them that you can't handle their dog along with yours and you need them to take care of this. If you can't get assistance at that point, I would recommend going to the homeowner's association (if applicable). Then proceed to the humane society, then police if necessary.
You don't have to make an enemy, just explain what you are observing, and why you need it stopped. Good luck!

First I want to say that Beagles are very very hard to keep confined unless you chain them up or cage them.
My dad has a few of them and we have one. Ours climbed OVER the chainlink we have along the back of our yard. My dad has a couple of his kenneled and a couple chained (he lives on a farm so he gives them chances to run as well)and the ones in the kennels climbed out like ours did. He had to build a roof over them.

Ok, other than that, when our dog does get out I would wander the neighborhood yelling for her, so it was obvious I was looking for her. She also has the vet's number on her tag. We had to resort to chaining her when we weren't out there or when she wasn't in the house. I would ask the neighbors FIRST. I believe it's illegal to just give away someone else's dog. I feel for the pup if they aren't giving her all the attention she needs but Beagles are high strung and love to escape. Just let the neighbors know you are not happy with the situation.

M.,
I want you to know that I HAVE THE SAME THING HAPPENING TO US! We have a basset hound that cannot stay out of our yard. She has overcome every obstacle that we have tried to prevent her from coming in the yard. We talked to our neighbors and we have both agreed that she is in love with my German Shepherd and they will pretty much do anything to see each other. If the dog coming into your yard plays with your dogs, then he is probably looking for the attention from either you or your dogs. I would first talk to your neighbors and mention that the dog is always over in your yard and see if they care. If they don't I would see what you can do as far as the law goes. You do not want to be held liable for something that may happen to the dog when he is in your yard or if he does something to someone else. I wouldn't take the dog unless you talk to a police officer about the matter first and the mistreatment of the animal can be proven. Take care!

Yes. I would talk to the neighbors right away. I wouldn't offer solutions to them, as it is their problem. I would ask them to take responsibility for it. I would be firm, but kind....."This is NOT okay."

If I were you, I might ask my vet his opinion on the situation. Or call the Humane Society to get advice. You could also just mention to the neighbor what the dog is doing, and see if they respond with compassion towards the dog (maybe they just don't realize they are neglecting him and may perhaps change their behavior towards the dog) or with defensiveness (in which case authorities may need to be contacted.)

Just a thought.

Does the dog get along with your dogs? I agree with the person who said the dogt is lonely. I would talk to the neighbor to make sure they know it is going on. *I* wouldn't do anything else. If they are feeding, etc. the dog and your dogs get along fine and they agree they are responsible for *anything* their dog does or for anything that happens to their dog as a result of coming out of their yard, I would let it go and let the dog have the companionship it apparently is desperate for. I would also check with law about who would be resposible for the dog and his actions in this case.