Richard Simmons can stay in hiding, but his new dog would still have to get out and do its biz, right?

Richard's longtime housekeeper, Teresa Reveles, was spotted outside his house walking a pooch. We don't know if it's her pet or Richard's ... but it would be awesome if it was his.

You'll recall Richard's beloved dalmatian, Hattie, died right around the time of his last public sighting 3 years ago. People close to him believed it was a big part of his depression.

If the new pup is Richard's, it could be a sign he's bouncing back. As we've reported ... police recently went to his house for a welfare check, and we're told he's just fine.

Richard's camp says the woman pictured is not Reveles, but another housekeeper who works for Simmons.