While configuring post to pulse in case level ,there is no option to mention as private post or public post . Let me know if there is any way to mention public or private post inside the case level and we are using version 8.1
Are you using pxContextFeed section for pulse UI ? If yes, this is old and you should use pxFeed section instead. Also, please check post and private post options in pxFeed parameter configuration. You should see a drop down on pulse post screen.