Within Teamwork Chat, you may find that you are unable to chat to other users when you initially log in as an external user.
As an external company user you cannot write to anyone initially. It is only when a user in the owner company writes to you directly that you can reply.
Once someone in the owner company adds you to a conversation or channel, you can send messages in that conversation forever as long as you are a member.
Note: As an external user, you cannot see other members of your own company in the people tab within the sidebar. Once someone in the owner company adds them to a conversation, they can then send messages in that conversation.
For more information, see: Enabling and Chatting to External Company Users