The publish notification goes out to users who have subscribed to the "Game Schedule Published" or "Practice Schedule Published" notifications found in the user's Settings > Notifications.

It is a one time deal.  Once the school publishes the schedule and the notification has gone out - that is it.  We wouldn't want to send people extra notifications if it was toggled on and off again - so in a situation where this was set to Yes and saved multiple times, the notification will still only go out once.

At this time, we do not have a way to track that this notification was sent, and to which users, however, we do have an existing request for this in our Ideas Portal here