To hide a member's contact information on the website and/or their directory profile, start by going to the record in the admin.
Then go to the Settings tab, and toggle the appropriate checkbox:
Hide on Website - this hides the entire record on the front-end
Hide Contact Information on Website - this will still show the record on the front-end but hide their contact information.
Hide Address on Website - this hides only the address on the front-end
Note: This is an admin function only. Members cannot do this themselves from the front-end.