I'm reviving this idea because I think it would be handy to be able to leave private messages for people. Perhaps the messages could appear both in-game (in a person's mailbox in front of their house) and they could get an email notification with the full message (if they choose that in their settings) so they don't have to log into the game to see it if it's not convenient at the moment.
Reba, this already exists as mail in the game :) Look for the big blue boxes to send mail (usually found near residential streets), and for skinny silver mailboxes for checking/receiving mail (usually found on residential streets).
edit: Just be sure that you've added the person you want to send mail to.
@Millie - yes, I already know that, but thank you for replying with what would be helpful info if I didn't know! :-D
However, I'm very interested in sending players messages when the game is closed, or even after it's 24/7 when I'm not "in the game" per se. I just made a very loooooooong reply to someone on their profile page and I think it will be more convenient to be able to send private messages to someone's profile than in game only and I thought this thread was suggesting just that.