skype works on the hdx. calls and such. the only issue i'm having so far is sending messages. this is because when the keyboard pops up, it essentially covers the enter button.... and the enter button ont he keypad just is used to enter a line in your message (doesn't work as a send button). it's very annoying. otherwise, i haven't had any other issues.
Not sure if it is my HDX or skype, but my camera looks right on my page but it is flipped upside down for who ever I am talking with. So I flip the kindle putting the camera at the buttom and it works but such a bad place for the camera to be. I can't find any thing on this.
I also have sound problems but fully turning it off and starting it up seems to fix it mostly. If I don't they sound like a robot.