Hi,
my 2 Cent:
I would not work on one set of images/ database from 2 computers. My past experience (especially with the also from you mentioned move/deletions) tell, that this is quite a safe way into confusion.
I would rather determine what is the “leading” installation for normal edits and only occasionally move out images when I want to edit them on a different machine and then re-import back into the main database.
Not sure, but with darktable here also the “local copy” feature might be of use?
→ “Local copies” in the darktable usermanual
You should also keep in mind that editing on two different screens with potentially different color calibrations might bring issues when finalizing images for print/display.