I agree with dmac66.
All of the posts on this thread are either about theory, or telling you to let it go. I am bummed out at those responses.
What I would do is firstly verbalize the obvious and say something like,
"cut that shit out. You aren't doing yr dog any good and if the right person sees you abusing the helpless, you are going to get beat the fuck down".
Or, something less escalatory at first.
That may be enough to embarrass this person into stopping.
But I don't know, whoever beat's their dog is on a totally different level than anyone I know, so maybe not.
If they start to get all punk on you and are like "it's my fucking dog I'll kick it if I want.",
well the you are at a crossroads. If you are confident in yourself and want to beat the tar out of this person, I think you have more than enough justification.
Someone who beats somebody who loves them and can't defend themselves is the worst type of shithead there is. They are the type to rape and hit their partners also. Ideally, everyone who acted like this would get the beating of their life, and their dog taken away.
If it's a recourring problem with someone you know, and you don't want to handle it yourself, get a crew together and take care of it as a team.
tl;dr: No-one should be allowed to walk away after physically abusing a loved one, especially one who can't fight back.