When I purchased my land some 15 years ago.....I had two neighbors with dogs that were running loose on several occasions. One was to the North of my land and one to the South. In both cases, I went to the owners and nicely told them of my problem with their dogs running loose on my land....and the problem I had with that interfering with the deer......and that was the purpose to which I had bought the property.
When I bought the land I sent a "nice" letter to each of the abutting land owners with my name and contact information.....and told them the purpose I had for the land (hunting and habitat development). Told them I would respect their land and hoped they would do the same....and to contact me at once if I were infringing on there property in any way. <------that went a LONG way with any of my new neighbors I talked to after that.....and they kinda welcomed me to "the hood".
Fast forward now to the dog issues.........
The north dog owner said "say no more, I will take care of it" and he did. (he had mastiffs)
The south dog owner said that they had always let there dogs run that land when Potlatch owned it. I said that they no longer owned it....and I have a different purpose than they did. She said "I understand".....and that was the end of the problem.......tho their horses did get loose a time or two and ate much of my brassica.....lol. (downed fence due to logging on their place). (they had some pointer-like and retriever dogs)
Occasionally just talking about it will work. But the trick is to stay calm and don't get into demanding.....especially when you are the new kid on the block. If anything, bring them a stick of deer sausage or something. You can catch more flies with molasses than vinegar. I think my original letter went a long way with both parties.
I have not had any dog problems in recent years.....knock on wood.