I've noticed, although I can't say with 100% certainty, but I had about a 5 Watt difference in 20 min power when switching between shoes.
The lower reading came with a pair of Sidi Energy, heavy, nylon sole etc etc and the higher power output with a pair of DMT Radial, so, stiffer, carbon sole, lighter.
I don't think the weight had anything to do with it, probably the stiffness, if anything. I'm using a Quarq though, so I wouldn't see any drivetrain losses which are probably relevant in your case as well.
Do you suffer more when you train, or cannot train?