Although some carriers will pick up outgoing mail in an apartment building if you leave it where they can see it and indicate it clearly.
1. Remove all mail from the mailbox
2. Place outgoing mail in the mailbox
3. Raise the outgoing mail flag
4. Carrier will empty the box and file outgoing mail during the next delivery before putting new mail in the box
Edit: here is a good example of a mailbox with that red flag https://www.amazon.com/Step2-541200-MailMaster-Mailbox-Black...
But this way around it makes a lot more sense.
Like they always shove (or drop) it into the mailbox so when there's letters that are obstructing the box from closing properly they know they didn't place them there so it's outgoing and they grab it.