Most people believe there is no way to view private YouTube channels, other than having the owner allow you permission to view the private videos.
However, it's kind of ironic that Youtube seems to be full of hacks and tricks that will guide you on the path to viewing private Youtube videos:
How to view a private Youtube video according to Youtubers
Unfortunately, I am not convinced - and I still think the only way a private Youtube video can be seen by anyone other than the owner of the channel is:
1. If they un-list the video and make it public
2. If their account gets hacked (but that would be a different question altogether: http://technology.blurtit.com/1149820/how-to-hack-someones-youtube-password)