Can you? A site's TOS will say that by sharing your data, you grant them the right to display, reuse and redistribute it, the same as you do now. And that would take precedence because your host provided the data. They requested and you provided.

The only thing that would change that is actual legislation. But then the legislation is orthogonal to personal data storage. If you want legislation for that, pursue legislation for that. Personal data storage is completely separate, and the two shouldn't be confused with each other.