My experience was with a new game DVD that had two scratches; one was on the surface of the shiny side, the other was a really nasty zigzag scratch that was embedded in the plastic (the game’s title screen wouldn’t even load).
I didn’t want to go through the hassle of returns from Amazon Marketplace sellers just for a very cheap game, so I set about fixing the problem myself. There was nothing that could be done about the embedded scratch, but I thought maybe it would work OK if I fixed the surface scratch.
I tried toothpaste, 2 different brands of furniture polish and car wax. The toothpaste was totally ineffective; it did nothing to improve the original scratch, and succeeded only in introducing dozens more tiny scratches. Furniture polish fared slightly better - there was some improvement in the appearance of the original scratch, and it didn’t introduce any more scratches. Car wax was the best; it almost totally eliminated the original scratch, as well as the vast majority of the small scratches caused by the toothpaste. Alas, the embedded scratch was the problem, so the game still didn’t work.
One thing I would suggest is not to try buffing it out too many times. I tried each of the above substances 2 - 3 times (so maybe 9 or 10 attempts in total). After all this, the two layers of plastic that make up the DVD were beginning to peel apart, so the disc had to be trashed anyway.