Can I affect how students receive notifications?

Not Directly. Students select what they are notified about, and how, through their personal notification settings.

When something becomes available to students (for example when you enter a grade for a student), they will be notified about it. If you do not want a student to be notified of the grade being entered (for example when you want to release the grades for all students at the same time), you should hide that grade item before entering grades, and then make that grade visible when you want students to be notified about it.

See: Grades

Similarly, if you have an important message that you want to ensure that all your students receive, you may want to send an email via the Classlist tool rather than using the announcements tool to post a message. Students have the ability to disable notifications from announcements or notifications from an entire course site, which means they may not receive a message and you may not reach every student.

See: Classlist