When publishing an event I chose to make this Private using the following setting: Published (Private): Does not display on the Constituent Calendar, only invitees with the Event URL can view the event. On our site's Calendar however this event is still listed and appears to be public.
The functionality to set a calendar event to "Private" instead of "Active" so that it is published but unlisted only works if you have the new calendar layout turned on. The layout is not turned on by default for clients. Contact support to request to have it turned on.