>>715717427Discs store data in the shiny foil part, the plastic itself is there for structure, but it needs to be transparent so the laser can shine and bounce on the foil. If the scratches aren't too deep, polishing until its level will fix read errors (Provided the only issue with your disc is scratches).
Polishing compounds are basically just super fine grit sandpaper, and toothpaste is a polishing compound.
The issue is that when polishing anything, you want to start with a coarse grit to get rid of the deepest scratches, then move to a finer one in order to remove scratches caused by the previous one, so like
>>715718010 said, it's gonna take forever to polish with just toothpaste. However, plastic is still fairly forgiving when it comes to skipping grits, 800 grit, then 2000 and then polishing compound usually doesn't take too long. Either that or buy a polishing machine for like 40 bucks on Amazon, they only use polishing compound but they're automated so it doesn't matter if it takes a long time
TL;DR
It works but it's likely gonna take you forever, you can speed it up via sanding with 800 and 2000 grit prior, or automate it with a machine that costs like 40 bucks