The "right of privacy" isn't.
That is, it isn't a right to do what you choose to do in private without undue busybody meddling by others.
"It", to be clear, is the legal right of privacy found by the Warren/Douglas court to be emanating, as a postmodern penumbra, from the U.S. Constitution in the contraception case Griswold vs. Connecticut.
In the comments of a post below Lydia asks:
What if we really were talking about purely private acts that you would have to go nosing around in other people's lives to find out about?
It is a great question, and it illustrates the facile nature of the claim that the delightfully postmodern "right of privacy" is about private acts.
Griswold vs. Connecticut wasn't about Griswold's private acts. It was about Griswold being able to buy his pack of publicly-marketed rubbers and strut around with them in public. If his acts were private we wouldn't even know about them.
It isn't "private" unless you keep it in the closet. Once you make it public it is, well, public.