If they read the general terms of the site they know its saved. Its really just common sense and being over thought. This law as a whole has so many flaws in it that nothing is ever going to be 100% compatible.
Case in point, you fill out a form on my site, it stores the form contents, then notifies me through Gmail. Now I have just shared your information with a 3rd party. Do you also need a notice to cover the notification of personal data being sent to you?