If a player would draw a card, that player exiles the top card of one of their opponents' libraries face down instead.
Each player may look at cards they exiled with Shared Fate, and they may play lands and cast spells from among those cards.
Each Shared Fate tracks only the cards it exiled. If the Shared Fate which was responsible for a card being exiled leaves the battlefield, putting another Shared Fate onto the battlefield will not allow you to play that card again.