At the bottom of the message box, right below the Send Message button, there is a checkbox, beside which says, "Save a copy in my outbox." Make sure there is a check in that box, and you'll have a copy saved.
You can either check the box every time or go through your profile.
If you go in your profile and click on Personal Message Options, you'll find a box that says Save a copy of each Personal Message in my outbox by default. Click on that box and then save your profile.
Hope that helps!