Even unregistered anonymous users can send messages to registered users of your website. This is possible only from the Send message web part, the My messages web part can be used only by registered users.

 

To allow this functionality, go to properties of the Send message web part and check the Allow anonymous users check-box. By checking the Allow anonymous users to select recipient check-box below, you can give anonymous users permission to view a list of all registered users and select a user from this list.