Public - Public projects are accessible to all users registered in the Mantis. Means all users who can access the project can see this issue.
Private - means that users with access level higher or equal to a specific threshold (typically DEVELOPER) can view the issue. The reporter of the issue will also see it, independent of their access level. Simply Private issues are accessible to users with a specific access level to the relevant project.
When creating a project, by default projects are public. You can even select private directly in View Status field, as shown in below image.
View Status - whether the project is viewable by all users that can log in to your Mantis (public) or private. Private projects are only visible to users who are assigned to it or to administrators.
To mark the project as private by default, you need to set the following config option in "Manage - Manage Configuration - Configuration Report"
All Projects, All Users, default_bug_view_status, integer, 50