For this test you'll need two browsers, ie Firefox and Chrome, and three users.
Make sure each user has at least one message to and from each other user. Make them something easy to identify. For example, "hello user A, this is from user B."
Make sure one of the users is your admin.
Pick one of the other two to be your target user. They must be enrolled in a course. Make sure that user has the other non-admin user as a contact.
As admin go into a course and click on participants in the navigation block. Click on the name of your target user.
You'll notice that the participants part of the navigation tree now includes the user you selected. Click on "messages" under your selected user.
The target user's contact list should be displayed. Click on one of their contacts and check that the correct users are displayed.
Note that the add/remove contact icon etc are hidden as you are not allowed to do that stuff while viewing someone else's messages. Search is also unavailable as one user searching another user's messages is not currently supported.
Open your other browser and log in as the target user. Go to their messages. Click on the same contact.
The two screens should now match aside from the small differences noted above.
In both browsers select the various options available in the messaging navigation drop down and check that the stuff displayed is the same.