BuddyPress.org

If you have a look at the roadmap you can see that in the next major version we can expect fine grained privacy control.

I believe Jeff, one of the moderators here, is working on a privacy component, that might make it into core. I don’t know when he will release it, though.

In the meantime, you can copy registration/register.php from your parent theme to your child theme and add a sentence that profile pages are publicly accessible, so it’s not a good idea to save any credit card details…