I never studied this, but a quick look at the
Transverse Doppler Effect on wikipedia shows what's going on.
Basically, the earth is moving through space in an orbit around the sun, which means it isn't necessarily going to be in line with the axis of the original direction the beacon was released in. Hence the Doppler effect will be different.
This picture may help (from the wikipedia article):
The left hand image is the reference frame of the beacon. The right hand image is the frame of the observer, so this is the one that explains the question you're dealing with. What it shows is how the wave fronts are actually spherical, so if the observer is at an angle to the motion, the doppler shift will be different.
In the image, the little black dot on the right is the observer. It's pretty obvious that as the beacon passes the observer, the compact wave fronts at the front of the beacon quickly become spread out at the back. It's the same basic concept as when a car zooms by and makes the classed 'nneeeyowwwmmm' noise. The car is constantly moving in that situation but there's still some point at which the doppler effect is zero, at the midpoint where it goes from 'neeee' to 'yooowwwwm'.
So for any situation where the motions is not on the same axis, there's always going to be a variation in redshift as the angle between the observer and the object varies.
So this is how the observation angle can change the amount of doppler shift, and explains why there is some angle where the shift = 1.