I would probably catch errors as you would on server side logging, log them according to severity and have the ability to turn on/off logging of certain types

I would probably catch errors as you would on server side logging, log them according to severity and have the ability to turn on/off logging of certain types on errors via config (eg. log.debug, log.warning, log.fatal etc). Although I've never had client errors logged to the server, I have had it logged in a hidden "console" (uses Firebug if available, otherwise custom logger). This gave us enough info to debug the error and we could tell users to open the console and copy paste the trace (was accessible via key combo Ctrl+`).