Does Time Machine run backups even when no account is logged on?
Solution 1:
Time Machine is a system-level daemon, not a user-level process. Time Machine will work even if you are not logged into your Mac.
Solution 2:
No. Mac OS X doesn't mount external drives so even if backupd was told to back up from an ssh session using tmutil startbackup the external drive that Time Machine normally uses would be absent and the backup would fail.