Hi Heather,
I notice you’re running version 4.3.3 of The Events Calendar. We are currently on version 4.3.4.2, so please update to the most recent version. Once that’s done, please review the Testing for conflicts guide and reply back with the results.
I did a quick test to see if I could reproduce the behavior, but events marked with the ‘Privately Published‘ status do not show in the mini calendar widget when viewing as a logged-out user. (see screenshots below)
Logged in:

Logged out:

Let me know what you learn after updating and retesting and we’ll go from there. Cheers!