The Chromecast has been evolving rapidly over the last few months. After getting a final SDK and a whole boatload of new apps, Google has now added support for private YouTube videos. Yes, you can cast any private video you've got to the Chromecast. Embedded videos gained support back in February as well.
Google has updated its Chromecast support page to reflect the change in functionality. We've tested to make sure the new feature is live, and everything seems to be working as expected. Keep in mind you are still unable to play live content or videos not approved for mobile – no way to know when or if those restrictions will be done away with.
This should work without any updates to your Chromecast or Android app. It's apparently something Google changed on its end, so it carries over to private videos you are viewing on desktop and mobile.