Hi,
this is possible with our Social Network plugin (demo at http://socialnetwork.easy-profile.com/), but this is only one function, it add many other features. Keep in mind that you can't use only conversation function, this make Easy Profile as a Social Network.

Another way is to use some other component, for example for UddeIM (this is very popular and free) you can add messages features to your Joomla site. For UddeIM there is also a plugin that add a button to send email to displayed user (this is free in our download area).