In a 2 player game, One of the wasteland encounters early in the game involves finding a corpse and finding a picture on said corpse with a name. Then, you add 005 from the library and shuffle it to the top of the settlement deck. (The 005 card continues on the story of who Margret is when you see her in a market). My problem\question is now the next player goes and draws the 005 card from the settlement deck which expands on the story that the previous player just started.

How would the next players character even know who Margret is? It seems like the 005 card should only be drawn by the player that inserted it into the game. Are we playing this wrong somehow?

You are playing it correctly. It's a common problem with co-op games. Eldritch Horror does the exact same thing.

I like to think the other guy heard about it on the radio or something like that.