A couples session is one appointment, billed once. It is not charged to each partner separately. For insurance, the session is usually claimed under the plan of one partner, who is recorded as the client, and the receipt is issued in that person's name.
If both partners have benefits, coordination of benefits can help. After the first plan pays its share, the second plan may cover some of the remainder, particularly where each partner is listed as a dependant on the other's plan. Your insurers can tell you how they coordinate.