The default setting is for projects to be created as public by default. Public projects are accessible to all users registered in the MantisHub. In order to mark a project as private which only makes the project available for administrators and users that are added explicitly, you can add the following config:
Go to "Manage - Manage Configuration - Configuration Report" and adding the following config:
All Projects, All Users, default_project_view_status, integer, 50
This will take effect for all newly created projects.