I thought about this some more, and it came to me a few minutes after I posted the bounty. :)
This isn't necessarily a full proof way of doing it as there might be other errors going on in Site.com not captured in the debug log.
When debugging a Force.com Site you can set up a debug log on the Site's Guest User. I would think you can do that with Site.com as well, because it has a Guest User as well.
If your Site Name is "Test Site" go to the Debug Log page and type in Test Site and the Lookup icon. This should find you the Guest User. Alternatively, if you have access to the Profile you can get the User's name directly. In Site.com you can get to the Profile through the Site Configuration page. Once you get to the profile detail page click the View Users button.