No, each would observe that the other's clocks had slowed down. So it would always be, unless the ship decelerates, turns around, and accelerates toward earth again. Then you have a situation where the ship's clocks have been through some motions that earth had not experienced. This gets complex, and I'll rely on Physicist to go into detail (as I always do when I'm over my head). When the ship gets back to earth, it becomes obvious that it's only the ship's clocks that had slowed down.
Hmm, again. I see where you're going with this, but then we have to ask the question of why the twins' paradox happens, since the two clocks would simply sinc up again when our hypothetical ship flew back towards the earth and both parties observed the other's speeding up.