Up to and including Exchange version 2010, a property check box can be available to set this up but it is much easier to issue a cmdlet.
Exchange 2010/2013 and Microsoft 365
Set-CalendarProcessing -identity <room> -RemovePrivateProperty $false
Set-CalendarProcessing -identity email@example.com -RemovePrivateProperty $false
Set-MailboxCalendarSettings -identity <room> -RemovePrivateProperty $false
Set-MailboxCalendarSettings -identity Liberty@door-tablet.com -RemovePrivateProperty $false
Manual Setting (Exchange, Microsoft 365, Domino, G Suite)
From Door Tablet V8.1 you can set words so that if they appear in the meeting subject, the meeting will be regarded as private. The words must appear either at the beginning of the meeting subject, or at the end. The text is not case sensitive. Private meetings do not show the subject and the organiser name is blanked out.
To set the privacy words, open the system profile, add your private words, delimited by semi-colons.
From version 8.4.12 you may mark meeting spaces so that all meetings taking place in those meeting spaces are private.