They're private tweets, the only way someone will get that info is by hacking your twitter account. Your bank account isn't ever known, that's tied into your twitter name on the banks system. Make a special private twitter account that doesn't have any personal info at all on it, if it gets hacked nobody knows who's it is, you create another twitter account and go to the bank to switch it over. It's just an easy way to get your info without having to be on the internet, just sms a message and get it sent back to your phone in seconds.

If your twitter account gets hacked you just call up the bank and disable the twitter feature till you get the problem fixed or a new account?