How can I avoid permission issues when syncing files across multiple Linux systems?

I am using dropbox to sync between 8 different Linux machines. I use BasKet notebooks to keep track of projects, and like many other applications, it has sync issues due to permissions requiring "do not sync" this and such file: permission denied. There are many tutorials on how to fix this temporarily using the hidden "Allow Permissions" button in the interface or with chmod, but the problem arises because umask creates permissions that cannot be synced across systems.

What is the best practice for solving this problem? Is there another system, like Isync with Google Drive, OwnCloud or Ubuntu One, that would be better than Dropbox in this case? Do I just need to host my Dropbox on a FAT partition? Any guidance would be greatly appreciated, thanks!