Does it also include the beta sites that are still in the private stage?
Yes, they are licensed under CC-BY-SA. If you go to the timeline page for a post you can even see the current license version:
The CC BY-SA says "You are free to share — copy and redistribute the material in any medium or format". Including not public?
What you can do
You can share CC-BY-SA licensed content in a non-public matter. For instance, you could post a copy of any Stack Exchange post on a private corporate intranet, print a book with your favorite questions and give it to your friends (and several people have done this), etc.
If you wanted to, you could (assuming you comply with the attribution and sharealike requirements) create a site that mirrors the content of a private beta site. It wouldn't accomplish much, since the sites aren't very private anyways — anyone can see the content on a private beta site by creating an account.
Note that that the quote in your question isn't from the license itself, but the human-readable summary, which is not a substitute for the license. The actual license text should be consulted for understanding what exactly you can and can't do. The grant of license is:
Subject to the terms and conditions of this Public License, the Licensor hereby grants You a worldwide, royalty-free, non-sublicensable, non-exclusive, irrevocable license to exercise the Licensed Rights in the Licensed Material to:
A. reproduce and Share the Licensed Material, in whole or in part;
B. produce, reproduce, and Share Adapted Material.
There are no restrictions in that section—or anywhere else in the license—that disallow private sharing.
A note on DRM
Perhaps you think sections 2(a)(5)(C) and 3(b)(3) disallow private redistribution? The "restrictions" disallowed in those section are referring to Effective Technological Measures, not non-public distribution: essentially, it would disallow you from modifying SE content and then sharing it (publicly or privately) but using DRM (digital restrictions management) to try to prevent people from resharing your modified version. But assuming you don't use DRM to restrict access to your redistributed SE content (I can't imagine why you would want to do that), those sections aren't relevant.