At least in the US, regardless of what the official rules might be (I've never checked) you only need a return address if you want the item to come back to you in the event that delivery fails. If you omit it and drop it in a public box with the correct postage it will reliably show up.

This makes sense because the only realistic way to enforce a non-falsified return address would be to verify government issued ID when accepting the parcel.

It's an interesting point thought that receiving messages does require a physical address. I guess the analogy would be needing to log in to browse the message board but not to post? That would be quite strange.