In addition to Ravi's comments:
Lesson (mod_lesson) doesn't provide other notifications than 'notify student of marked/graded submission in essay question'. There are no Lesson module notifications a teacher can set up other than maybe setting up an event monitoring rule (but I wouldn't recommend that).
Assignment has a variety of notification settings, which include 'notify when work submitted' and 'notify when work submitted late'.
For all things related to receiving notifications, the cron maintenance script should be up and running and an
SMTP server should be set up. If you don't have direct access to those settings, please consult your Moodle's administrator. If both are working correctly, be sure that the mail server isn't treated as a source of
spam mails and that there aren't any thresholds (max number of emails) getting in the way of processing the notifications.