How to turn on and off certain fields in the People profile page?

This has to be done by App Manager.

To turn on/off visible fields in People profile:

1. Click on your avatar at the upper right hand corner, click on "Manage Applications".
2. Go to "People" > "Profile Fields".
3. Check the box next to the fields you want to be visible on the People profile. If you make a field visible, you can select whether you allow user to edit the field.

All visible fields are searchable.

Please note that if you synchronize that field with Salesforce or Workday, that field will not be editable by users.