the card holder, presumably the same person here, would get the bill at their real address in their real name...authorized users do not get separate bills, they all go the the original card holder
say i setup my step-son, who has different last name than me, as an authorized user on my card while he is away at, say, Sam Houston College, while I live in, say, Podunk...his signature and name - even picture - are on the card, but i receive and pay the bills