I get a few volunteers each year. It seems I can never find all of the tubers when harvesting. It is easy to move them into a row rather than have them scattered. Yes, they will produce.
I never put tater peelings nor any part of a potato from a store back into the garden or compost heap. It is simply trying to avoid introducing disease.
If my potatoes grew without disease last year, I will save some seed (tubers) to plant this year. If there was any disease last year, I would opt for certified seed potatoes this year. I have had good luck in my current location avoiding potato diseases. No other gardeners right near may be a help too?