Do observers at different speeds perceive other speeds differently? I was told that if a plane takes less time to travel from the China to US as opposed to the other direction due to rotation of the earth. I suspect this is incorrect however. 
From this scenario I came up with the following thought experiment. Imagine two people sitting on a train. One person is sitting with his back (A) facing the front of the train and the other is facing toward the front of the train (B). They are throwing a ball to one another. They pass by an observer who is standing outside as the train passes (C). How does each person view the ball?
My intuition tells me that (A) and (B) should perceive the speed of the ball the same while (C) should see the ball move faster towards (B) than (A). However some people have told me that time taken for the ball to reach (B) when thrown by (A) should be less than the time taken for the ball to reach (A) when thrown by (B) due to the relative velocity of the speed of the ball and the train. Is this true?
 A: The flight time from the USA to China may be different to the return trip, but if so I'd gues this is due to the jet stream rather than the rotation of the Earth.
Anyhow, your intuition about the train is correct. The two passengers A and B see the A to B and B to A speeds of the ball to be the same, while an external observer sees them to differ by twice the speed of the train.
The situation you describe is an example of Galilean invariance. This can be surprisingly hard for non-phycisists to get their heads round. After all it took mankind about 2000 years (from the first Greek scientists until Galileo) to understand it. This is the way I look at it:
Suppose you're passenger A in the train, and we also need to assume the train is travelling at a constant speed - this is important because if you're in a train you only feels any force when the train is speeding up or slowing down. You throw the ball to B at e.g. 5 m/sec. If the train is moving at constant speed the ball also moves at a constant speed, so it reaches B at 5 m/sec. B now throws the ball back to you at 5 m/sec, and again the ball doesn't change speed so it reaches you at 5 m/sec. So the speed is the same in both directions.
The key bit is that once you've thrown the ball it doesn't speed up or slow down during it's flight. An object, like the ball, can only speed up or slow down if there is some external force acting on it, but once the ball is in flight there is no force trying to speed it up or slow it down (we're ignoring air resistance).
Things get more interesting if the train is accelerating, but that's another question.
