What's the order of resolution for "everybody discard" in MTG?
You go in AP-NAP order around the table to choose a card.  However, the actual act of discarding is considered simultaneous, which means the chosen cards are still in the hand, a hidden zone, until all are chosen.  It is perfectly fine to, for example, set the chosen card aside face down until everyone has chosen one.


  101.4. If multiple players would make choices and/or take actions at the same time, the active player (the player whose turn it is) makes
  any choices required, then the next player in turn order (usually the
  player seated to the active player‘s left) makes any choices required,
  followed by the remaining nonactive players in turn order. Then the
  actions happen simultaneously. This rule is often referred to as the
  ―Active Player, Nonactive Player (APNAP) order‖ rule. Example: A card
  reads “Each player sacrifices a creature.” First, the active player
  chooses a creature he or she controls. Then each of the nonactive
  players, in turn order, chooses a creature he or she controls. Then
  all creatures chosen this way are sacrificed simultaneously.
  
  101.4a If an effect has each player choose a card in a hidden zone, such as his or her hand or library, those cards may remain face down
  as they‘re chosen. However, each player must clearly indicate which
  face-down card he or she is choosing.