MCC Online users receive message alerts for a number of different reasons and can also send and receive instant messages. How a user receives notifications and messages from their courses can be controlled in Settings > My profile > Messaging.
This brings up a screen with check boxes where users can choose how they wish to be notified both when logged in and logged out. They can choose between a pop up or an email, or even an Alert badge notification (that bell icon at the top right of your screen).
To see more details, check out the information at moodle.org: http://docs.moodle.org/22/en/Messaging_settings