The app should give you the option of which contacts and SMSs should not be sync'd from the phone. For instance, I never want one-time PIN SMSs being sync'd due to obvious security concerns. Ideally, I'd like to be able to configure exclusions based on message content partial matches.
This is a very good suggestion.
This could also be solved, I think, by deleting messages in the web app that are deleted on the phone. (Make it an option for those who would like to do this).
YES! I WAS ABOUT TO WRITE THE SAME THING! I am uninstalling this app, however awesome it may be, precisely because my bank uses one-time PIN SMS for authentication.