If a student shares an address with one or more parents, as here on a contact card:

Two shared Addresses

Then the address will 'nest' under the 'Show Additional Information' button, under the parent's name in the directory:

Directory: Nested Addresses

However, if a student has an address that is not shared with parents, as here on a contact card:
(in this example, the 'unshared duplicate' address is not flagged with 'Mailing Address' to avoid duplicate mailings)
Contact Card: Unshared duplicate address

Then this unshared address will show up separately in the directory, underneath the student's name, as here:

Directory: Unshared Address