Keep in mind, all speed is relative
in relativity. If there is one inertial frame (inertial = non-accelerating, constant speed and direction) where the Earth is at rest and the ship is moving at 0.99 the speed of light, then there is another inertial frame where the ship is at rest and it's the Earth that's moving at 0.99c, and both frames are equally valid in relativity. In the first frame it will be the ship's clocks that are running slow due to time dilation, but in the second frame it will be the Earth's clocks that are running slow due to time dilation, so as long as they continue to move inertially there is no objective truth about who is experiencing less time. On the other hand, if one of them changes velocities so that they are now moving towards each other rather than away from each other, then this is non-inertial motion so the symmetry is broken, and whoever it was who accelerated will be the one who has experienced less total time over the course of the journey when they reunite and compare clocks at the same location (this is what Hurkyl was talking about, different paths through spacetime between the event of the rocket departing and the event of the rocket returning have different total 'proper time' along them).
If you were traveling away from the Earth at 0.99c, and if the messages were sent once per hour in the Earth frame, then the time between your receiving the messages would be greater
than an hour according to your own clock (and if the messages were live video streams, everything on Earth would seem to be going in slow motion). Likewise if you were sending messages once per hour to the Earth, the time between your messages being received by people on Earth would also be greater than an hour according to their clocks--the situation would be totally symmetrical. This slowdown would not be due to time dilation alone, it would also be magnified by the Doppler effect (basically, if someone is traveling away from you and sending signals at a constant rate, then each successive signal is sent from a greater distance so it takes some extra time to reach you, so even without time dilation there would be some slowdown, although time dilation adds to the slowdown).
Then if you turned around and were traveling back towards the Earth at 0.99c, now you would each receive each other's messages at a speeded-up rate, because of the Doppler effect (now each successive signal is sent from a smaller distance, so it has less distance to cover to reach you than the previous one), which shortens the time between signals being received by a greater factor than time dilation lengthens it. So, in this sense things are symmetrical during the inward journey too. There is an asymmetry over the entire journey though: if you spent the same amount of time traveling away from the Earth as traveling back towards it (same speed on both the outward and inward legs of the journey), then you would see the Earth's signals go from slowed-down to sped-up exactly halfway through the journey, while the Earth would see your signals slowed-down through most of the journey and sped-up for less than half of it, which is one way of explaining why you have aged less (sent less total signals to Earth than they sent to you) by the time you return. This is explained in the Doppler Shift Analysis
section of this excellent page on the twin paradox
, and in the Too Many Analyses
section they also offer a nice spacetime diagram, with time on the vertical axis and space on the horizontal axis, the signals sent from Stella (the rocket traveler) to Terence (the guy on Earth) shown on the left, and the signals sent from Terence to Stella shown on the right.
You can see on the left side that Terence is receiving slowed-down (redshifted) signals for more than half the time between Stella leaving and Stella returning, whereas the right side shows that Stella is receiving slowed-down signals for exactly the same amount of time she is receiving sped-up (blueshifted) signals (for her the switch from one to the other happens at the midpoint of her journey when she turns around, the curved section of her worldline in the diagram).