With macOS Catalina (10.15), Apple introduced new security and privacy features for your Mac.
In order to capture screen shorts and screen recordings with Dropshare, you have to grant it access to the Screen Recording permission.
Capture a new screen shot with Dropshare and you will see the following message:
Click "Open System Preferences" and allow Dropshare to access the Screen Recording permission.
You might have to authenticate first by clicking the lock icon in the bottom left corner of the window.
Afterwards, you are ready to go!