Kim, they should receive confirmation emails at the account email or email associated with their account in the LMS. Rarely I have seen issues with emails not being received but usually it's just that the submission wasn't completed and so there was no confirmation to send.
I actually have a different case -- my student received a confirmation email, but the submission was not found in my assignment inbox. She discovered that only when she was preparing for her next submission. What could've gone wrong?
Ana Chupungco If you are using a Blackboard LMS, check the Turnitin assignment inbox to see if it just didn't sync to the grade center. Otherwise, Turnitin can track it down using the submission ID from the receipt. I think in the 10 years I've worked with Turnitin I saw one submission mysteriously disappear and be retrieved by TII support, all other cases were unfortunately "doctored" submission receipts created by the student.