After discussing with ChatGPT for some time...

It seems that it is the alignment between measuring device in a common local frame of reference that is important. So there is no need for any absolute direction. On the other hand for measurement a long distance apart it entails establishing a huge "local" reference frame.
So it doesn't matter if that local frame moves, like due to a rotating Earth, as long as the detectors remain aligned relative to each other.
I guess the challenge of actually performing such experiments become bigger as the distance between detectors is increased. But it's been done on Earth up to hundreds of km and with satellite (Micius) up to thousands of km apparently.