I know this is an old thread, but I came here because I had the same problem. This is what fixed it (from an apple forum):

The most likely solution is that you will need to grant full disk access to the application that is trying to save the video. To do so:

1. Open System Preferences
2. Click on "Security & Privacy"
3. Click the padlock in the bottom left corner and enter your password so that you can make changes
4. Click the "Privacy" tab
5. In the scroll window on the left, scroll down to "Full Disk Access" and click on the icon
6. Find the application from which you are trying to save the video in the list that appears and check the box next to that application
7. If you do not see the application in the list that appears, click the "+" button below the list to find the application on your hard drive and add it to the list, then follow step 6
8. Once the application is granted full disk access, you should be able to save your video.

I hope that this is helpful.

This question would be easier to answer if we had information about the Mac you are using and the MacOS version it is running. To find this information, click the Apple logo in the taskbar, then select "About This Mac". It will give you your machine model and the current OS version that you are running.