Alright, so here was the deal. I had already set the permissions for all users to read/write, and that was what was shown on the virtual disk image file.
However, based on your positive feedback, I knew it must be possible, so I took another go at it.
It turns out that when you apply permissions to a directory, it updates the permissions of the files, but not necessarily the package contents, if it is a package. The disk image was a package, so I did a "view package contents" and then selected all and updated the permission to read/write for the individual files.
Now it all works fine.
Thanks!