There were normally two trays or boxes for paper on people's desk, one was the "inbox" and one was the "outbox," and the mail person would place the new stuff in the inbox and take anything from the outbox to circulate it around the office.
I never really lived that, although my first desk job did have an inbox and an outbox, the team wasn't big enough to have a mailroom, so we would just use those boxes to drop papers at other people's desks - hence the outbox didn't really get used, but the inbox did. Usually once a day a bundle of mail would get dropped off at the front desk and someone would take a turn walking the letters around the office, but there weren't that many of them as email was already a thing at that point.