My husband is a letter carrier, and once when he was collecting mail out of one of those big blue boxes, he found some loose checks made out to various places. He drove out to the woman’s address (it was local) and dropped off the checks, and she ended up sending a very nice letter to his supervisor.
That being said, sometimes the postal inspectors will run “stings” with loose cash, etc., on postal workers, so depending on what’s been accidentally/intentionally put into one of those boxes, you might not get as prompt of a response.
Every library system I’ve known will accept returns from other affiliated libraries - I can return books from the Thomas Jefferson branch of the Fairfax County Public Library at the John Marshall branch, for example.
During my public library worker days, we would often get returns for the library system in the next county over or for the college library. While our official policy was that we didn’t accept those returns, we often got them in our night bins. We would generally return the books to the appropriate library for the patron, but it was generally done in a “Jenny, you live over by the York County library - can you drop these off there?” kind of way rather than by any formal arrangement. As such, there was always a delay of at least a few days before the books made it back to their homes.