A target that is co-speed will typically have a doppler shift indeed. :)
I can use a couple of examples to explain. (Assuming look down and doppler filtering enabled)
If you are flying at 400kts transmitting at 10000Mhz, your radar will actually send signals at a slightly higher frequency due to doppler shift. For this example let's say that your current speed equates to a doppler shift of 1Mhx (exagerated).
If you then have an approaching target going at 400kts you will recieve an echo with a "blue shifted" 10002Mhz freq. Ground clutter would be at 10001Mhz and filtered out by the doppler filter. (since we know what speed we are going, we know what doppler we introduce ourselves)
If the target instead would be beaming you, i.e. going at 90 degrees to your heading right in front of you with zero radial speed. The echo would be non-shifted at 10001Mhz. This is the same freq as ground clutter, hence the radar would filter the target echo as clutter,
If on the other hand, the target would be co -speed, i.e. tail chase. The echo would be red-shifted at 1Mhz. So echo would be comming back to you at 10000Mhz. This is not the same as ground clutter and would then not be filtered out as ground clutter (we still know what speed we have and what doppler we introduce to the signal)
However, since there are alot more things going on, like side lobe clutter and general signal processing, radars will typically have weak spots in other areas. Above is very generalized to describe doppler. Thge newer the radar, the more tricks it can pull off in terms of mitigating these weaknesses.
In this example i've actually left out that there is a doppler shift when the echo is recieved by the radar as well... but it doesn't matter in this case.
So it basically boils down to the fact that doppler is always introduced to a signal if a transmitter, target or reciever has a speed component in the signal propagation path. But since we are looking for the target and not ourselves we need the target to have a speed component in the signal propagation path, i.e. it needs to be going either towards you our away from you.
All of the above is very simplified but I think it serves the purpose of explaining doppler in radars.
Hope it clarifies things a bit. :)
N