I removed Skitch with AppCleaner, reinstalled from the App Store, but Skitch did not request screen recording capability, nor can I find a way to add it. Skitch is currently not usable for me on a freshly installed load of Catalina.
I had the same issue after recently swapping computers for work. It's likely that you downloaded Skitch and it's operating from the default download directory instead of Applications.
Close Skitch
Open your finder and locate where the Skitch app is
Drag the icon into Applications
Relaunch the Skitch program from Applications
After this I was able to take screenshots again, even without setting permissions.
Screen Snap--> Capture not working
in Skitch for Mac Issues
Posted
I had the same issue after recently swapping computers for work. It's likely that you downloaded Skitch and it's operating from the default download directory instead of Applications.
After this I was able to take screenshots again, even without setting permissions.