Hey Steve,
That is a great question there.
There is no built-in busy/public feature in our plugins, but it could be done with some custom development and template override.
For template overrides check out our Themer’s Guide – https://theeventscalendar.com/knowledgebase/themers-guide/
For the month view I would create an override for this file:
wp-content/plugins/the-events-calendar/src/views/month/single-event.php
For signaling busy/free you could either use a custom field or a category.
Then, in the above file (in the template override) you can do a check if the event has the specific custom field and value or if it is in the specific category and based on that either display the event title or just “Busy”.
You can create template overrides for the other views as well, if needed.
Does this help you get started?
Andras