What they said! And whatever you do, never store an unripen mango in the refrigerator. That will stop the ripening process and you might as well make pickled green mango. Once it's ripe, it can be refrigerated up to a few days.
If the mangos seem to ripen only on one side, try rotating them everyday. But if you put it in a paper bag, you probably won't have to flig it.
FYIW, for a papaya, it should be set on its tip (upside down on the narrow end) at room temperature to help the ripening process. I typically put it in a glass to keep it standing. And while I'm at it, a pineapple should be also be stored upside down, although a pineapple will not ripen after being picked. Standing it upside down will redistribute the sugar and prevent fermentation on the bottom. It'll still turn yellow but it won't develop any more sugar. That's why it's very important to pick a ripe one at the store.