LibraryH3lp is designed from the ground up to be a multi-operator system. That is, you can have any number of concurrent operators staffing your service.
All operators with available (or chatty) status staffing a given queue will receive each incoming chat. Only the first operator to respond will become connected with the guest. The operators that do not "win" the chat will become disconnected from the guest, and they will receive notification of the operator that successfully answered the chat. All chats can be transferred to other operators or queues at any time.
In the following chat, amytest answered the chat first. The operator actually looking at this chat window did not answer it.
You can also use LibraryH3lp as your in-house chat solution. That is, create user accounts for all your staff (even if they don't need to chat with patrons). Then everyone on staff can chat with each other one-on-one or via group conference rooms.