Once, in a mad rush to grate some cheese, one of our colleagues popped some mozzarella cheese out of its package and right into the chute of the food processor - without noticing that the plastic nutritional information label was still stuck to back side of the cheese. We didn’t think he had the patience or skill to pick out every single shred of label, but apparently - given enough time - he managed it.

Now, with regard to your ginger, do you wish to eat the peel? Do you wish to spend time picking little shreds of peel out of your grated ginger? If you are going to eat it (as opposed to using it only as a flavoring and then straining it out), you should peel before grating.