As other folk have said, only during an eclipse. More specifically, light doesn't really diffuse like it does in atmosphere, because there aren't any particles to bounce off of. Technically space isn't a perfect vacuum, as there's stray particles all over the place. It's hardly enough to matter.

It seems like the near side of the moon could theoretically get more light on average. When light travels through the fringe of the Earth's atmosphere, it will diffuse and refract a bit. To some extent, some of that light will end up hitting the moon when it otherwise wouldn't have had the Earth not been there.