Gold = Has passed all exams (read: mental and physical conditioning) Can be used to do certain errands. Illegal to murder (but all other Yukkuri hate them)
Silver = This means the yukkuri is a pet and the owner has paid a certain sum of money to insure that the yukkuri bearing this medal will not be murdered on whim. Otherwise, it can't be used to do anything else. Silver badges may be promoted to Gold badges if they pass the exam, but that's very hard to do as most Gold badge yukkuri are conditioned from birth to pass.
Bronze = It does nothing. Just like goggles. It's basically saying that you acknowledge that this yukkuri is really shitty, but you have paid a sum worth in pennies so that maybe... just maybe people could restrain from killing it on sight. But if that fails too, no hard feelings.