I managed to get it working. The solution was to change the permissions of /media/tamas/ (The ntfs file system is mounted to /media/tamas/NAS.) to allow the debian-transmission user to read and write it.
I would also like to note, that in other places I have seen people look at /etc/init.d/ to change start scripts. Ubuntu doesn't use that, and if I changed the user, transmission starts under, there it didn't actually change that. Ubuntu uses systemd to start services, and my transmission-daemon start script was located at /lib/systemd/system/transmission-daemon.service.
One other problem I had was that I could not use shell as the debian-transmission user, because of its configuration. Then I found the following command:
sudo su --shell /bin/bash --login debian-transmission
I found this very useful in debugging the problem.