The Assignment drop-down can be found on the top right corner of the Messaging Module and it may be used to assign or reassign a contact to a user. The dropdown will list all the available agents with their avatars. In addition to that, you can also search for a user using the search box.
Next to the Assignment drop-down, there is a button where the agent can mark a contact as Done or Pending. When a contact is Pending, they can be marked as Done. When a contact is marked done, they can be marked as Pending when necessary.