The mirrors would hopefully mimic the curvature of the moon and cover 1 full half of the moon (the entire side facing us). If we are capable of seeing the whole moon during full moon (seeing all light reflected from it) then why wouldn't we see all the light reflected from the mirrors?
The light is scattered because the surface is matt, and not glossy. So you see light reflected back from all over the illuminated parts. It's no longer a "beam" it's intensity drops inverse squarely with distance. The proportion of light reflected is called the albedo.
Also Mirrors are so fantastically more efficient at reflecting than moon dust it stands to reason that the new moonlight from the mirrors will be more powerful than 1/1000th of the power of sunlight.
I don't think moon dust is that poor a reflector. It's albedo is not that low. It's just that the reflected light is scattered light, so 1 million miles away from it, it looks relatively dull compared to the sunlight.
In appearance, the mirror coated moon would look mostly black and invisible, with a star reflection on it here and there, a little blue dot in the middle where the earth is reflected (most visible at dawn and dusk), and a very bright spot somewhere on it where it's reflecting the sun.
But the total light we'd get from that bright spot would be not much more than an ordinary full moon. I'd expect maybe 2 - 4 times the total energy we get from the full moon now.
edit: I've look up the figure. lunar albedo = 0.12, which means the mirror coated moon would give us 8x the full moon light, which would probably be like average street lighting

But I am still guessing the maths there.