I've left onions to overwinter a couple of times but only when I wanted to try collecting seeds. Since onions are biennials, what's going to happen is that the plants will soon put up a stalk and start flowering. Once it does, this central stalk is going to become hard / woody and ruin what's left of the onion. The bulb will probably not get any bigger or will shrink as this happens as all the plant's energy will be directed into the flower.

I'd probably pull them either now or as soon as you see the stalk start and use them soon since they won't keep too well. You can try and plant the store bought onion but the same should happen to it and I don't know if the seed will come true nor if it would even form a bulb in your zone.

Darnit - I knew just leaving them alone would be too easy! I pulled several up and it's exactly as you describe - the leaves are growing like mad and the bulbs are shrinking. Since I have no idea what they were, exactly, I won't take a chance on the seed, either. Onto the compost pile they go, and ditto with the store-bought onion. I've got plenty of new onions in the ground and need the garden space for other things. Thanks so much for your reply, makalu!

Tosser