Well here's the problem. If you suspect the dog is being abused, you have to go through the process of notifying a local animal control agency, and then mention that you are willing to take the dog in for your own. In the U.S., there's no such thing as self-help. You have to go through legal and bureaucratic processes to legally transfer property be it exchange of cash through a verbal/written contract, or through some sort of legal proceeding as a result of a settlement. In any case, do it right, and you will have the dog without any complication. You may also wish to draft up an affidavit saying that they are giving you this dog, otherwise it can be considered a theft crime, regardless of any hearsay.
+1 to this. If my dog went missing, I would immediately print out flyers and inform my neighbors. You've had the dog for a month and you live 5 houses down. They obviously weren't searching hard enough. And like booger said, they were so quick to give it back to you. If it was my dog, I would fight you for it!
I say give it back unless you feel based on the way the dog looked or acted that he was neglected or abused. It's difficult to say someone doesn't care because they didn't put the same effort into something as you would've. Especially since this family had kids and whatnot, there are 1,000 things going on for the mom at any given time and she may not have had a lot of time to look. Apparently they do care though, which should be enough to give it back.