How do I get full details of each person when joining tables I know this has to have a simple solution and I am not seeing it. 

I have a table, person, which contains all information about each person such as first name, last name, age, address.

I have a second table called consumer_client which contains a row containing person id for each consumer and client like this: 

id | consumer_person_id | client_person_id


What I want is to fetch both full names when I join person to client_consumer like this:

id | Joe Mama | Mike Smith


I cannot seem to get that with a simple JOIN syntax like

SELECT * FROM person JOIN client_consumer WHERE person.person_id = cc.person_id


as it always returns just the consumer_person_id and client_person_id along with other information in person. I want it replaced with the full name instead of person_id.
I think you are looking at this the wrong way around. You should be selecting from the client_consumer table and joining to the person table. This way the query is easily read and understood.

SELECT
    cc.id,
    consumer.full_name as consumer_full_name,
    client.full_name as client_full_name
FROM client_consumer cc
JOIN person consumer ON
    consumer.person_id = cc.consumer_person_id
JOIN person client ON
    client.person_id = cc.client_person_id