otto64, the plugin has a check for if the user is logged in or not on line 206-209 in chat-room.php in the plugin file. While I wasn't around when the plugin was originally developed, I am around now. I think that would be a good idea to have a toggle for. If you need to, you can amend those lines and remove them if you want your chat to show no matter what.
However note that I'm now going to put it in the roadmap to offer a setting for that and it will be toggled to hide by default. Once I get that update out the door, and you upgrade, you'll need to take a moment to uncheck the option. Won't be anything for a long process.