I was absent minded when replacing the wipers on my wife's 2012 Caravan and left the plastic protector on the pass side wiper. She wiped it a couple times and it left scratches deep enough to catch with my finger nail, which surprised me. Any chance of buffing them out? Since it's the pass side one, nothing is in direct line of driver's view, but it is annoying.

most likely not without some distortion.

Unfortunately , If you can feel with your fingernail the answer is no,but the scratch in the windshield is not from the plastic on the new blade ..... it’s from the old blade that you just took off 👍🏼

I sanded wiper scratches out of the windshield on my '72 Challenger years ago with a DA and 1500 grit paper. Used lots of water and did not let it sit in one spot. It turned out pretty good, but I believe it was just dumb luck.

When I polished a mark out of a windshield the view was distorted in that area. If it was a light mark then you may get away with it but a deeper scratch would take a lot of polishing.