To have screen sharing functionality, you do need to have the latest version of Skype (I'm using version 188.8.131.52). If you need to check if you have the latest version, open Skype, click on 'Help', then click on 'Check For Updates'. If there are updates, download and install them.
A couple of useful explanations about how to screen share in Skype (you may need to upgrade to the newest version of Skype, and this could be why it wasn't obvious where to click). The first 'tutorial' is "How to Share Your Screen with Skype" - image from that site below.
The second is "Screen sharing will change online teaching", image from that site below.
Hope these are useful, and see you online again soon.