They both have advantages and disadvantages.
Pivot type are nice because they are very simple to work on, spring rate, spring preload, or dampening is very easy to adjust, you can run counterclutches just like the fixed setup. The bad parts are the pivoting itself. It makes it harder to place all the parts, mount exhaust and carbs, and the float level on the carb changes with the pivot. To have a wider gear change ratio you have to make the front a larger diameter and it makes the clearances an issue with stock setups (think 110mm plate from Bidalot crashing into the driven pulley)
The advantages of the dual fixed are that they take up less space for the same gear ratio change, do not require so many moving engine parts, are usually torque sensing on the final drive (so when you are under hard acceleration it keeps it in a lower gear for longer, that's why the rear pulley turns as it slides in and out, [double side note: that's why the ball ramp plate on a stock moby has the ramps not directed toward the center but slightly off to the side, it was the same idea]) These setups become a lot hotter for some reason and on all modern scooters and even Derbis are forced fan cooled. It's more difficult to regear most dual fixed types because you have to change the final drives in the rear hub, and you usually only have a 1 choice (derbi is the obvious example along with other chain drive cvt).
As far as actual power deliver and efficiency they are both going to have the same issues. Plate clamping pressure, spring rates, roller weights, belt slippage and curvature of the belt (how tight of a curve it goes around) all affect how efficient the belt system is. Higher power requires higher plate clamping pressure to keep from slipping and higher gear ratio changes require wider and stiffer belts, which are harder to bend. I've read a lot about trying to make them more efficient from a modern standpoint but not much of that really applies to mopeds. And it all ends up as a give and take. But to answer your question, I do not know if there is much of a difference between the two as far as power deliver go, but as simple designs go, everything leans toward the dual fixed side.