If you're talking about someone on a spaceship and a stationary observer (or Earth), then the person on the spaceship traveling at a high speed (relative to the observer), is the one that experiences time dilation. His clock slows compared to the observer's clock. When he returns to Earth, his clock will be behind.
If you look at it from the point of the traveler, his clock stays at the same rate, and the clock on Earth has sped up.
The relative velocity being +v or -v depending on the reference frame does not mean that both experience the same time dilation.
What causes the dilation traveling through spacetime. Regardless of perspective, Earth is traveling through space at the typical Earth speed and time on Earth flows at typical Earth time. Regardless of perspective, the (near) lightspeed traveler is traveling through space faster than Earth, and is thus traveling through time slower than Earth is.
You're suggesting that the traveler can say "Hey, Earth is just moving away from me at v, so should have a slower clock when it gets back to me." just as people on Earth say "Hey, the traveler is just moving away from Earth at v, so should have a slower clock when he gets back to Earth.".
This is wrong. The actual travel through the fabric of spacetime, from a Universal (or outside, fixed) reference frame is what causes the dilation. Flipping reference frames doesn't make the relative effect go away or apply to the other side as well. Only the people on Earth can make that statement.
We've done this. We've flown atomic clocks and they've come back slow (or fast, depending on East/West travel). A plane clock and an Earth clock don't experience the same time dilation. Time dilation is caused by traveling though space, not traveling relative to an observer. The measurable effect of time dilation is always expressed as relative between the two parties - a traveler would think Earth's clock sped up and Earth would think the traveler's clock slowed down - or an outside observer. This is because we don't have a Universal reference frame that grids everything to the fabric of spacetime.
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