In Live Agent, it is possible for a Supervisor to send a private message to an Agent. What about the other way round? I'm not talking about the flag to request help, but an actual private back and forth chat.
You have to use the Pre Chat API for that, and add a form to your webpage. You find a Pre Chat Form Code Sample in the Live Agent Developer's Guide, including a "Chat Now" button for the website visitor to click. It will look like this without any formatting / CSS styling done:
Remember to allow access to the Pre Chat API in the Agent configuration of your Live Chat Deployment.