I wanted to update this as I was looking and could not find an answer that worked On OS X El Capitan (10.11.5). I started poking around and I found them in a different location.
~/Library/Application Support/com.apple.sharedfilelist/ com.apple.LSSharedFileList.FavoriteServers.sfl
- Copy from the old location.
- Paste to the new location.
- Change the owner (if needed)
- Right (CTRL) Click > Get Info > Sharing and Permissions > Unlock > select or add user (making sure it has read/write > Gear > "Make user the owner"
- You could also use the terminal with "sudo chown 'username' '~/Library/Application Support/com.apple.sharedfilelist/com.apple.LSSharedFileList.FavoriteServers.sfl' " making sure to replace 'username' with your actual username.
- Relaunch the Finder (might not be required).
In response to the original question, you may be able to put this file in a DropBox folder and run an alias to the Library location. Not sure how it would work with multiple owners and permissions on the file as I don't use dropbox and haven't tested it.