You can move fields to tabs set to not display on profile or set fields themselves to not display on profile. Formatname is used for public name output and what it outputs can be adjusted in CB > Configuration > General using the "Name Format" parameter.
Additionally you can use CB Privacy to completely control all tab and field privacy either by forcing privacy or allowing users themselves to set privacy via privacy controls. This is probably the best option since you mentioned wanting users to have the option to expose information. Privacy controls in CB Privacy can have defaults so you can set fields to be default private. Take a look at its add-ons page below.
www.joomlapolis.com/addons/cb-privacy