I'm in the process of migrating from Mojave (on the internal HDD) to Catalina (on an external SSD).
Under Catalina, if I try to go to "Security & Privacy" in System Preferences, it hangs, and becomes unresponsive until I Force-Quit it.
I've seen suggestions specific to other System Preferences panes hanging, under different macOS releases, but not specific to Catalina or to "Security & Privacy."
I finally found the time to try a "safe mode" boot on the Catalina side today. At least, I think it was a safe mode boot -- it didn't say anything about safe mode, the way it does for recovery mode. I held the shift key while the switchover was in progress, up to the point where it asked for my password (both load-source drives, i.e., both the internal HDD and the external SSD, are encrypted), and then held it for quite a while after giving it my password.
It asked for my password a second time, just before the desktop came up, and the secondary screen did not come up (thereby dumping a s---load of icons onto the main screen desktop).
Attempting to go to "security and privacy" in System Preferences still threw up a pinwheel mouse-pointer. While it was doing that, I tried Disk First Aid. It was then that I realized that whatever was locking up System Preferences was also eating up lots of CPU time: performance was extremely sluggish just navigating to Disk Utilities. Once Disk Utilities came up, it showed both "SanDisk SSD" and "SanDisk SSD - Data"; I ran Disk First Aid on both of them. It appears to have found nothing.
A symptom I failed to mention before, but which may be related
When shutting down on the Catalina side (and this has been happening for about as long as the System Preferences lockup, again only on the Catalina side), Firefox will refuse to shut down unless forced. In fact, as I recall, if I'm on the Catalina side, Firefox will come up without my explicitly launching it. Even if I explicitly closed it before shutting down. Could this be a piece of the puzzle?