You can access your privacy settings by going under the settings of the Raider IO page, clicking on a character on the left hand side, and then clicking the Privacy link that will show under it.
Here you can make the following changes:
Show/hide your score in game; This will stop the addon from displaying your main score on your alts or score breakdown by spec. Since blizzard now always displays your score in the game, it will not prevent that.
Show/hide your raid progress in game.
Anonymize Profile References: Instead of showing your characters on the leaderboards and in runs, you will be given a randomly generated name that starts with "Anon."
Profile Visibility: You can allow your characters page to show or be unviewable.
There is also a check box here that will allow you to apply all the changes you made to all your other characters.