keslo

var Game = function (players) {
  this.game = {
    players: []
  };

  console.log(this.getStateWithMe(players[0]).players); // тут все нормально
  console.log(this.getStateWithMe(players[1]).players); // и тут вроде тоже
}

Game.prototype = {
  getStateWithMe: function (player) {

    var g = {};
    g.players = this.game.players.slice(0); // копируем массив-значение из this.game.players
    g.players.push(player);

    return g;
  }
}

new Game(['a', 'b']);