If an operator closes the chat, the chat remains open on the client side and allows input from them. There is no way that I can see to re-open the chat from the Visitors page as it's gone once the operator clicks the close chat button. From looking at the chat history you can verify the visitor can still type.
The same problem exists if the chat is taken over and the person that takes over the chat closes the chat. The first operator can see the client's input, but has no way to take back the chat even if they have the permission.
Is there a way to close the visitor's side of the chat to stop taking more input and insert a generic message to designate the chat is closed?