My main question is how do I aproach my neighbors without sounding like an a$$hole.
Dave, a very valid question. Some dog owners may not realize that it is going on. I would suggest telling them politely that their dog is indeed running loose, and include if you saw the dog actually trailing deer.
If the neighbors don`t care, or worse show an attitude about it, I would pursue EVERY legal action against them.
Many good points have been made concerning the reasons NOT to shoot a dog. I blame the owner, and have no problem doing what I must to end the problem ASAP.
BTW, back in the 70`s and early 80`s, the PA game commission said it was your duty as a licensed hunter to shoot any dog witnessed running deer.
People who allow their pets to run wild are not on my christmas card list, and I have no problem telling them graphically!