What is the distinction between "'private' videos" which are presumably created by individuals, and privately owned media (like movies) which are created by corporations?
Having the video encrypted (since Netflix is pushing for it anyways, why not use it to my benefit), allows me to provide the keys directly to the content owner for decryption.
I guess the built in support for basic encryption would work here (the mandatory profile).
You could even deliver the symmetric key out-of-band with asymmetric encryption.