Grr, turns out that they were using a "share" on the folder. There is a "General" share, which the people at base use and I was using at base and there is a direct "Project" share, which those on the remote site use, because they don't usually have access to the whole drive.
Although the directories had the correct rights assigned for each group, the Share was set to "everyone, read only"!
So, although the user had rights to see write to the directory, the share was restricting them to reading them only.