I have it on mine - I think I downloaded the app for the Kindle Tablet - doesn't work with the video in the UK as we are not on 4G which is a disappointment and it also doesn't seem to like headphones. Other than that it works well.
Amazon use to show the Skype icon on their Kindle fire page, funnily it got removed after I'd bought to kindle fires for my daughters as gifts. Now I find myself half way across the world with two frustrated little girls back home unable to chat with Daddy as they wanted. Very bad show Amazon, and really skype what is so wrong with giving us access?