There is no clear and easy way to see private messages. If you no longer have the email with the quick link, you have to first find the user who sent you the private message (you have to remember who it was!), navigate to their profile and then click the "send private message" button. Only way to do this in 1.6.3.
Now, if you give 1.7-beta-2 a try you'll see there is a new "Private messages" menu item in the profile section. From there you can see the messages you've sent and the ones you've received.
That was from the core's perspective. Involving some plugins, you can take a look at https://github.com/q2apro/q2apro-on-site-notifications
which seems to have added private message notifications.