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The thumbnail generation for private videos on Vimeo does not load the images because the thumbnail is loaded via Simple API instead of the Advanced API that loads the images via oEmbed.
As quoted from vimeo for the same issue in Media:Vimeo in http://drupal.org/node/1587686:
The Simple API is for public data only. If your videos are private, you will need to use the Advanced API. If your video is set to hide from vimeo.com, but is embeddable, you may also be able to use oEmbed.
Attached is a patch that use the Advanced API for image loading.