I was googling "joomla messaging component", and this forum post actually showed up 4th for me. I understand it's generally not good practice to respond in threads so old (this one started in 2010), but because I found it and the answer didn't seem 100% clear, I'll post a quick response.
I just recently wrote a tutorial on the Joomla Messaging Component
and in my testing I noticed that you can only send private messages to Administrators and Super Users.
There are a few other things to keep in mind as well:
1. You may be able to update the components options to allow yourself to send messages to registered users (I'm not 100% on this, but according to the settings it looks possible).
2. When you send anyone a private message, they get an email notification of the new message. Still, this email notification does not include the message itself.
3. In order to read the new private message, the user HAS to log into the admin dashboard for joomla. So, if you allow the component to send registered users messages, you must also allow them to log into the dashboard (which *may* be a security issue).
Please let me know if I can help any further.
Thanks, BradM, with http://www.inmotionhosting.com