Yes. I would absolutely love to have this.

Dude...Yes.
I hate it when I send a message and don't save it, then lose information or forgot what I sent.
Def. yes.

Yes could be helpful.

This already exists,

There is a option to

Save message in Sent Private Messages.

Exact same thing in my idea, my vote is No

This idea is no good and I'll tell you why. There is a little check box on the bottom of EVERY message that says, "Save a Copy of this Message in my Sent folder". If you don't have the sense or the inclination to click a little box then you don't get to save the message. So in other words there already is an outbox and its called the "Sent Copies" folder. Simple answer.

Already exists.

Check the Save a Copy of this Message in my Sent folder checkbox before sending your message. This will not be automatically on, as a vast majority of users forget to empty their sent folder.