"Private listening" is a feature available by plugging headphones into your Roku remote or with your mobile phone connected to your Roku with the Roku mobile app. Any audio that is played on a Roku will be redirected to your headphones if connected by either method (not both at the same time). I don't think there is a way to access it any other way. It is not something that developers have access to. depending on the Roku model it may be a blutooth or wifi direct connection to the remote. I have not encountered any documentation that indicates what transfer method the stream is sent as.
https://support.roku.com/en-gb/article/217478308
https://support.roku.com/en-gb/article/217479488
What would you like to accomplish that you think using "Private listening" would be the means to accomplish your intended use case?