Answer
Each observer sees a rainbow that reflects from a different set of rain droplets.
Work Step by Step
A rainbow is not a physical, tangible object. As described on page 530, the rainbow you see is from droplets that lie within a 3-D cone, with your eye at the tip.
If you slide your eye over to coincide with that of a nearby observer, the imaginary cone moves with you. The light from the new rainbow is coming from a different set of droplets.
It is in this sense that two observers standing apart do not see the "same" rainbow.
