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?
Both ways are correct and dependent on who is playing the game and what their skill and preferences are:
When players are getting used to the game they are encouraged to play with their hands face-up to simplify the communication and memory aspect of the game.
To increase the importance of accurate communication and memory and to give everyone something to contribute to the group plan, the players can play with their hands face-down.
From the official rules, p.7
When playing the Introductory game (4 Epidemic cards), place your cards face up in front of you, for all players to see.
- Only Player cards count towards your hand limit. Your Role and your Reference cards are not part of your hand.
When playing the Standard (5 Epidemics) or Heroic (6 Epidemics) games, keep your cards private, so everyone has information to contribute to play discussions.
- Experienced groups may decide to play with open hands, if desired, in these games.
Players may freely examine either discard pile at any time
Pandemic Legacy should always be played with cards face up. There are mechanical reasons for playing this way.
In Season 1, there will be effects that trigger as soon as a particular card enters someone's hand. Playing face up all the time makes sure that people don't forget these triggers (even unintentionally).
in Pandemic Legacy, players keep their hands faceup at all times
Players keep their hands face up at all times.
Normal Pandemic is generally played with hands face down, but you can choose to play with hands face up if you prefer. The reason for face down is usually to prevent one player from having all the information and trying to analyze everyone's turns and tell them what to do. You can choose to share whatever information you want, but you aren't required to do so.