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.
Even better, the developers worked in a different building to our users, so there would often be a couple of days lag to getting a response. If you needed a quicker answer, you had to phone them, or schedule a meeting if you needed to explain something in more detail.
We used stacks of "endless paper" core dumps / assembler listings when painting rooms, as you could just unfold a huge stack of paper to protect the floor from paint splatters.
I suspect that one of the reasons "starting in the mail room" used to be so venerated is that it's a good way to get to know a wide swath of people inside a company in a relatively short period.
Needless to say, the mail-room were privy to a lot of information should they care for it.
Edit: Working where both emails and physical memos were used: Drafting a physical memo elicited a very different state of mind from banging out an email. Printing it out, proof reading on paper, there was a lot more care via reflection knowing this would be in someone's hand and they'd likely be looking at it on their desk, giving full attention, making notes with pen over it, rather than simply a message displayed on a screen while multiple other tasks pop up from other incoming mails or calendar reminders or IM chats. Slower, but a level of preciseness and everything needing to be just right, all the way down to how many spaces after a period vs a colon.
They were completely interchangeable with emails, including in the way they were written (like short letters). I don't understand why we used them, but alas, we did.
It wasn’t used that much, though, except for the odd journal subscription that someone had delivered to their office address. The system would only work with envelopes and magazines, anything bigger had to be collected from a central mailroom.
Sending mail between offices was a little more convoluted —you could just use the post or a courier yourself, but if you got your document or memo in the hands of the right person (typically a mailroom clerk) before a cut-off time, it’d be sent as part of the company’s larger delivery.
[1] https://en.wikipedia.org/wiki/Mail_chute
PS. Google Image Search has a good selection of office memos.
If I have a package delivered, there's someone with corporate security clearances that will swing it by the desk.