Private Projects are created under the folder My private projects. They can only be viewed by the owner, so these Projects cannot be shared. However, if the owner joins a Group that prevents members from having private Projects, any private projects are moved into the Group Projects folder. This will allow the administrator(s) of the Group to view any private Projects.
Shareable Projects are created or moved to the folder Group Projects. They are visible by default to the Group administrator(s) and can be shared among all team members by changing its share settings.
In the advanced Labfolder version, Group settings can be configured so that private Projects are not allowed. If this setting is active, all private Projects of newly invited Group members are moved into the respective Group folder. However, you can copy the content of a private Project to share with members of a Group.
Tip: You may also want to read the article dealing with the closely related question of why private Projects have disappeared, explaining what happens to private Projects when joining a Group preventing them by its Group settings.