I have a 2019 16" MBP running Monterey and a 12.9" iPad Pro running iOS 15.1. I upgraded from Big Sur yesterday, and before then SideCar was functioning fine. After Monterey completed, SideCar would connect to the iPad but it would be a black screen.
I did uninstall my Cisco VPN client software and then rebooted, but then SideCar would work only sometimes, and for a few minutes before disconnecting due to bad network quality. Other times the screen would freeze with a static image and no longer update.
What I had forgotten is that the trust relationships between devices often get cleared out/need to be refreshed after OS upgrades, so I plugged in the iPad directly to my MBP with a USB-C cable, told the iPad to trust the computer, and it's been working for me for about five minutes or so at this point. Maybe it's working now?
[EDIT] If I try to start up SideCar with an Elektron Overbridge plugin running it goes back to the old behavior, and if it isn't, I can get that behavior dragging the window back and forth between the MBP and iPad. If I run the Overbridge plugin while SideCar is running, I see a rust-colored frame with black cut-outs. It is routing audio, from the device to the host, however.