I finally worked it out. You need to create a public/private key pair and then add these to the backup server's authorised keys within the profile of the user you want to log in as. I'd been doing this, but not as root and that was why the transfer was falling over - shame the debugging info wasn't that great.
Login to the backup server and create the account for the backup.
sudo adduser rbackupNow log into the host server, create keys and enable passwordless login to the remote backup server - remember not to specify a passphrase for the key pair.
sudo -sConfirm the login process by ssh email@example.com
ssh-keygen -t rsa
ssh-copy-id -i /root/.ssh/id_rsa.pub firstname.lastname@example.org
If you don't now create the directory that tbackup is expecting to log into remotely, it fails with a very unhelpful message. So, if (in the settings section of the tbackup page within Plesk), you have set backups to be stored in /rbackup/foo, you need to ensure that his exists on the remote system.
If there are issues with tbackup and you need to do some debugging: