I had uninvited tenants coming to my house to pay the rent so that it wasn't late and it was a PITA. I got a PO Box and I LOVE it. The paranoid tenant can hand it to the postal clerk at your post office and have him/her put it directly in your box. This way the tenants don't know where you live and you won't have any surprise visits.
If this is truly a PO Box, what you describe is illegal. The postal clerk cannot simply place unposted uncancelled mail into your box. And in my experience, they will not do so.
I have a mail slot in my garage door, under a security light which automatically comes on at dusk. I've operated like this for 7 years and have never been bothered by tenants when they drop their payments into the slot. I don't answer the door and I instruct them that if it goes anywhere but the slot, it isn't received.
My PO used to have separate slots for local and out of town mail. The local mail was postmarked at the PO and then delivered directly.
Since the USPS went to the large processing centers the local mail slot has disappeared. I think they now have to send it 2 hours away for processing so it can be returned to the local PO for final delivery. This process is supposed to be much more efficient. ;)
Now to date myself....when I was a child you didn't need to even put a stamp on local mail. Local postmasters had the power to decide whether or not to require one and Gib had decided that we didn't need one.....times change....