Private posts, or "announcements," are generally not visible publicly; instead, they're displayed on the "Announcement" element of a portal page that's only accessible to members of a certain role or group. Role/group members have to login to access the portal page; the page itself is not available to public users. This is also why announcements don't have RSS feed URLs and are not displayed on News pages - both of those would represent ways that public users would be able to read private announcements.
It is possible, however, to display announcements on a News Archive page. Create the page in Page Manager, and select the "Show Portal Announcements" checkbox.
Users will still have to login to the site in order to view announcements, and they will only be able to see announcements targeted at the groups or roles they belong to.