Resource Links Broken When Synced to Second Computer

Mark’s point 2 is the cause of your problem, you may think that by having the files synced via BitTorrent or Dropbox to identically named folders means they are identical … They are not … Not as far as OS X is concerned.
OS X gives each file a unique invisible name/tag/label (whatever they call it) and its that tag that RW is linking too.
So 1 file synced between 2 machines has 2 unique tags and thus any RW references you setup on one machine is broken on the 2nd.

You can test this very easily:
Create a file in any app and save it to a folder
Close the app
Move the file to another folder
Open the app
Open the file from the recent list in the file menu

It WILL open the relocated file. This is because the recent file list uses the invisible tags not the drive/folder/file path to store it’s info.