A teacher was getting a Private Access Only error when the students would click on Youtube links.
Schoology has confirmed that this is a bug.
The workaround is to use the “Share” link at the bottom of the video instead of copying that URL from the top of your browser.
Visit the video you want in youtube and then click on Share under the video. Copy URL and then paste into Schoology.
Even using this method, Schoology still did not like displaying the video inside of Schoology so I had to uncheck the View In Schoology icon when editing your video.