To begin with YouTube will upload your videos with private as the default setting. This means only you can view them. Unfortunately you only get the “This video is private” message once you attempt to watch it.
This is also true if you embed a private video on WordPress. YouTube will let you capture the embed code and publish the video to WordPress where it will look like it has published correctly but when you click to play it you will get a “This video is private” message.
Always check that your videos play even if they look right.
The screen cast below explains how to change the privacy settings on YouTube.