Says Microsoft,
Skype Screen sharing Black screen
If you face this issue, create a system restore point first and then see if any of these suggestions help you:
1] Update Skype
Update your installed Skype software
2] Restart screen-sharing
Click Stop Sharing. Then again Start Sharing again. If you have a secondary monitor, then use it and see.
3] Update or rollback Graphics drivers
If you recently updated your graphics drives, roll them back; else update them and see if it helps.
4] Reset Skype
Open Settings and Reset the Skype app and see.
5] Edit the Registry
You could disable (VbSS) and enable Remote Desktop Protocol (RDP) and see if that helps. Open Registry Editor and navigate to the following key:
Create a new DWORD, name it EnableConferenceScreenSharingOverVideo and set its value to 0Also create a new DWORD, name it EnableP2PScreenSharingOverVideo and set its value to 0
Next, navigate to the following key:
Create a new DWORD, name it EnableConferenceScreenSharingOverVideo and set its value to 0Also create a new DWORD, name it EnableP2PScreenSharingOverVideo and set its value to 0
Restart your computer and check. If it does not help, you may reverse the changes or go back to the created Restore Point.