By default, users will automatically receive a normal email when they have a new message. 99% of users choose not to disable notifications. Once setting are changed to disable the only way to see notifications is to log into the courses and see them from the course dashboard.
Start by selecting the instructor drop down (see image below) from the instructor home page.
From the drop down select Account (see image below)
Select MY ACCOUNT and from the drop down select EDIT PROFILE (See image below)
From the Edit Account Info page you can do a lot of things. By default, the boxes are checked for users to get an email for private messages. Users can uncheck the box and they will not get email from the system.
It should be noted that once they opt out of getting these emails the only way they will know if there are messages is to log into the system.