I just found out that you can visit the project page of a project that is private. Here's an example (@bh, is it ok if I use this as an example?)
https://snap.berkeley.edu/project?username=bh&projectname=test
But you still can't view projects that don't exist
https://snap.berkeley.edu/project?username=bh&projectname=nice
The first link should go to the same 404 error page as the second link.