I followed this site:
https://help.ubuntu.com/community/SettingUpNFSHowTo
I set everything up as recommended.
But I was unable to change the permissions.
I was using
sudo rsync -azv
So, after a bit of googling I changed
/etc/exports
/export 192.168.1.0/24(rw,fsid=0,insecure,no_subtree_check,async) /export/backup 192.168.1.0/24(rw,nohide,insecure,no_subtree_check,async)
/export 192.168.1.0/24(rw,insecure,no_subtree_check,async,no_root_squash) /export/backup 192.168.1.0/24(rw,nohide,insecure,no_subtree_check,async,no_root_squash)
The key was
no_root_squash
Then I could read and write as root.