How do I get full details of each person when joining tables
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