We tend to use pearl sized onions (we grow them from scallions which end up turning into small bulb onions which are amazing). We store them in our pantry, keeping them cool and dry.There are people who report to us that they bought onions from us in the fall and were still using them in the spring (heck, we've done it too). If we do have leftovers (maybe from a salad), it will certainly go into the fridge. There has been no evidence that bulb onions produce any toxins/bacteria (unless of course contaminated). Onions are very acidic and therefore aren't quite so welcoming to bacteria.
We keep our potatoes in the pantry as well, cool and dry. They generally last a while. www.snopes.com/food/tainted/cutonions.asp<