in this Lindybeige video, he argues that the educated elite actually spoke greek in the privacy of their own homes instead of latin, but he provides no sources backing that claim. Now, I know that greek was a very important language for scholars, taders and diplomats, and that many people studied it, but is his claim well-founded, or is it just speculation?
Maybe. ;) This question was asked previously here: Did the Roman elite speak Greek to each other?. I think they did while appropriate, while /u/legalaction was a bit more hesitant.
In all honesty, we can't make a definite conclusion on the language of private conversations.