edit 2: graphs.facebook.com will aggressively and unfortunately normalize any YouTube link to youtube.com. See, for example, http://graph.facebook.com/http://www.youtube.com/watch?v=ZB8...
This give the number of shares for youtube.com, not this individual video. I am not sure how to get facebook share stats for a particular YouTube video.
Facebook and Twitter, do basic normalization, but do not unshorten shortened links.
edit: This has already been discussed on HN: http://news.ycombinator.com/item?id=577457
edit 2: graphs.facebook.com will aggressively and unfortunately normalize any YouTube link to youtube.com. See, for example, http://graph.facebook.com/http://www.youtube.com/watch?v=ZB8... This give the number of shares for youtube.com, not this individual video. I am not sure how to get facebook share stats for a particular YouTube video.