Imagine you have a light source at rest between two detectors which starts emitting light waves.
It would look something like this
Each light wave spreads out in a circle at c, with each wave front hitting each detector at a rate equal to the emission frequency.
Now consider what happens if the the light source is moving towards one detector and away for the other as seen from the frame of the detectors.
The light waves still spread out in a circle at c, but the origin point of each wave front is closer to one detector than it is from the other and the center of each circular wave front is displaced from the previous one. The wave fronts hitting the blue dot are closer together and the one hitting the red dot are further apart. since they are still traveling at c with respect to the detectors, the Blue dot see a higher frequency of light and the red dot a lower frequency.
As far as the light source is concerned, the waves still are moving outward from it at c in circles, but the red dot is running away from the wave fronts traveling in its direction and the blue dot is rushing towards the wave fronts in its direction. So according to the Light source frame we get the same result, wave fronts hit the blue dot at a faster rate than they hit the red dot.
So no matter which frame you pick, you expect the detectors to register Doppler shifts.