|You can't both have it open and in read/write mode at the same time. Make sure all other instances of the file are closed before opening it on the remote computer and then see if you can make changes in it. If you can, that was the problem.|
If that isn't the issue, then it's the permissions you've given the other user on that file. Go into the ACL on that file/folder and make sure you have set that user's permissions to read/write.