If your posts are presenting the above message to users that are not logged in (can test in a private browser window), there is an easy fix. In the post editor, look for the “Access” metabox within the editor sidebar. Make sure to change the value to “Everyone” from what it is currently set to and update your post. Now your post should be visible to everyone.
I ran into this feature accidentally, the previous post on this site ended up having the access set to “Anyone subscribed” instead of “Everyone”. This went unnoticed by me for several weeks until I ended up visiting while not logged in and noticed the ‘Subscribe to continue reading’ message. Of course, the Access metabox was minimized so it wasn’t immediately clear what was amiss.
Additional Thoughts
While the access feature is neat (if you want to use the various Jetpack features for managing subscribers), it somewhat clashes with the existing Visibility field for posts (which WordPress has had for nearly ever). It would be nice if this feature integrated as additional Visibility options (where it really belongs). Additionally, it would be good if, as with the visibility modes (private/password protected), there was an indication in the front-end that the post is not accessible to everyone while logged in (to help avoid surprises).
-John Havlik