Under Setup->Monitoring->Debug Logs you can configure debug logs for any user.
When click the magnifying glass in the screen shot to the left (in production) and search for site* or *user to locate any public site profile. If you have recently viewed said public site profile, then it will appear in the recently viewed user list as well.
As of today, you can't do either of those things in your sandbox instance.
If you go to the site and click "Public Access Settings" to view the sites profile, you can see the name of the profile. You would think you might be able to copy this name into the search box and add the user to the debug logs this way. That is not the case as the name displayed on that page isn't the actual name of the profile used by this screen.
So how do you add a debug log to the public site? Type in "Site" or "Site*", hit enter and it will either select the user (if you only have one site). If you have multiple sites defined, you will get an error that says "Multiple items found. Select from drop down or click icon to refine search" with a drop down to select the site.
The moral of the story is that the look up box you bring up by clicking the magnifying glass doesn't have the same behavior in the sand box as in production in that it won't search the public sites profiles. However using the search box on the main screen will.
I share this in case someone wants to debug code in the sandbox instead of their production system.