Whenever I try to use the colour picker on Sketch, it asks for Screen Recording permission, which I did allow. But it still asks for permission again if the app is restarted.
Hello Rajdeep 
The prompt you receive is linked to the control access screen recording feature that macOS triggeres everytime the you allow third-party apps or websites to record your screen.
Our apps use this feature because of the color picker feature. Now, from your report looks like this feature gets reseted everytime you restart your mac.
I will recommend you to:
- Go to System Preferences > Privacy and security > Screen Recording and check if you see the Sketch app in the list of authorized app
- Restart your Mac
- Make sure the preference are turned on once you reboot the computer (before opening any app). - System Preferences > Privacy and security > Screen Recording
If the app is not on the list after your restart, you may be experiencing a macOS bug requiring further assistance from the macOS developer.
On the web, if you quickly search Privacy & Security settings in macOS reset on reboot, you might find similar reports to your case.
Let me know if this helps. Cheers 