TL;DR

How does a frosted diffuser affect the amount and distribution of light passing through it?

One way to measure the total light output by a source is to use an integrating sphere. This is a spherical cavity coated with a Lambertian surface,

https://en.wikipedia.org/wiki/Lambertian_reflectance

Like your frosted led, this surface scatters light without significant absorption. The effect is to randomize the light causing each element of area to be of uniform in intensity independent of the directivity of the source. Measuring the intensity of light from a known area then gives the total light since the total area is known. Measuring an LED’s output before and after frosting might yield you an answer.

To a good approximation when light interacts with transparent plastic it’s either scattered or absorbed. Frosting is a roughening of the surface which shouldn’t change the material's absorption.

One way to measure the total light output by a source is to use an integrating sphere. This is a spherical cavity coated with a Lambertian surface,

https://en.wikipedia.org/wiki/Lambertian_reflectance

Like your frosted led, this surface scatters light without significant absorption. The effect is to randomize the light causing each element of area to be of uniform in intensity independent of the directivity of the source. Measuring the intensity of light from a known area then gives the total light since the total area is known. Measuring an LED’s output before and after frosting might yield you an answer.

An integrating sphere spreads the light from the source uniformly over the interior by multiple scatterings. The white sheet has just one scattering of the emitted light.

But an integrating sphere seems like a lot of work to me. If you have some very white Lambertian paper you could make an integrating box to experiment with.