Fired it up to check my own app (RunnerCast) and it looks fine. The Liquid Glass menu moves from the bottom to the right side automatically both on the inner and outer screens. If you already had a layout that worked for iOS and iPadOS, things adapt fine to the inside screen and when you go split screen with two apps, the layout adapts nicely. I don't think I'll need to do a thing.

I've also noticed a flurry of recent app updates from some of the bigger apps (MLB, NYTimes, etc.) to switch to the Liquid Glass menu which I assume Apple strongly suggested they do prior to the Duo launch.