Dogs on leashes (Town & USFS ordinance) are welcome on the Mammoth Nordic/Shady Rest trails. There is a doggie bag dispenser right at the trailhead! The problem isn't anywhere as bad as it used to be out there once certain dog owners (after they got cited for dogs off leash & not cleaning up pooh) began taking their pets out to where they stop plowing the Sherwin Creek Road near the propane tank farm beyond "The Station" (the meadow area off Old Mammoth Rd). You could take your friends' pooch out there, too.
Just watch out for brown klister!