it really depends on how your doing the backup in the first place.

for me. i run my backups via cron and i have the output goto the cronusers email. you can always pipe the output to the mail command and then you can specify which user.

some backup software allows you to configure users to be notified when an event happens. so i would check your software docs and see what you can come up with, or talk with the people that manage the backups and set it up in the first place they would prolly have the most valuable information for you.