I would like to be able to follow sites that have already started private beta, so that I get a notification in the mail when they are open to the public; as well as it appears in my profile.
Why is currently not possible?
The concept of "following" was really designed for a proposal. Once the site is launched (even though it's in private beta), the proposal is, essentially, closed. There's nothing left to follow.
You can now "follow" a proposal, even after it enters private beta. By following the proposal, you will be notified when the site enters public beta. We added text to the proposal page and the private-beta login page to let everyone know they can be notified.
[Follow] this proposal to receive an email when public beta begins in 6 days!