The answer we received from Salesforce Tech support is that before the Winter 17 release, if you turned on debugging for a Site Guest User, you might receive an avalanche of debug logs if many different people were accessing your Force.com site at the same time.
After the Winter 17 release, you now also need to set a server cookie in the browser to enable debug logs for Site Guest Users. This means that debug logs will only be generated for those doing the debugging, not everyone accessing the site. This is true for sandboxes also.
If you look in Chrome's Cookie settings you should now see a 'debug_logs' cookie, and so the next time you reload your page, debug logs should re-appear again.
There is more documentation on Debug Logs here, but at the time of this writing, adding the cookie is not documented:
Salesforce: Set up Debug Logging