In playing Pandemic, I've always played where the cards in your hand are face up - i.e., visible to everyone. It is a cooperative game after all, and looking at another player's cards is easier than asking everyone what they have each turn. Certain actions require you to have knowledge of other player's cards (e.g., Share Knowledge), so it doesn't seem to be prohibited that you do know what cards the other players have.
In watching a video of BoardGameGeekTV playing Pandemic (well, Pandemic Legacy), they keep their hands face down. The Pandemic episode of Tabletop had the players' hands face up. Which is the correct way to play?