Q. Why do some of my contacts have a community name displayed and others don’t?

A. The community name is displayed when there is a community selection made within the contact record (valid only for those who have community/locations) configured?

commnameCapture

If your communities are close together and a contact is shared between locations you may elect to not specify a community choice or depending on your organization preference you could have two contacts with unique community designations.

It would then be helpful to link the two together with relationships in order to see what communication has been distributed to referral contact.