I actually had the issue after a Skype update while using Mojave. In Mojave there was no such privacy setting (for screen recording), so I updated to Catalina and, while the setting was there and I set it for Skype, it didn't solve the problem. Since the bug exists for other non-microsoft screen sharing applications, my guess is that it's a bug that Apple introduced to application developers when they provided info on their new privacy/security settings, and screen sharing applications thus carried this bug over to their applications. Regardless, I think it will require fixes by both Apple and Microsoft (and other screen sharing applications).
I joined the "office insider" program to get early releases of updates and got a skype update today. It still didn't fix this bug.