Facebook Oauth and sending messages to facebook users
When a user registers or connects with his/her Facebook account, your web application receives that person's Facebook username which you can then store in a field.

On Facebook, you can receive external messages at username@facebook.com.

Now all that's left to for you to do is set up Rules module (maybe Facebook Rules module can help) so that if a user has connected his account on your website with his Facebook account an email is sent to username@facebook.com.