Answer1 told you how to set up a Trust. That is the answer to your question.
In the Domain drop down, you see the 2 different domains NOT because the individual workstation clients are members of 2 domains, you see them both because of the Trust.DOMAIN_A trusts DOMAIN_B and vica versa.
Clients connected to DOMAIN_A are ONLY connected to DOMAIN_A. If you opened AD Users & Computers in DOMAIN_B, you would NOT see the clients of DOMAIN_A listed.
The Trusts running between the Domain Controllers in DOMAIN_A and DOMAIN_B makes it so that users from EITHER domain can log into workstations in EITHER domain, but you would have to change the Domain drop down to the appropriate domain. So, user BOB belongs to DOMAIN_A. BOB goes to a machine that is a member of DOMAIN_B. Due to the Trust between the 2 domains, BOB can sit down and log in as BOB, but he must choose DOMAIN_A from the drop-down. If he left it at DOMAIN_B, he would not be able to log in (because BOB does not exist on DOMAIN_B).
hope this clears things up