There are actually two answers to this question, yes and no.
When an event is first created, if a participant is added to it, the software is programmed to automatically change the event title to the participant's name. For most teachers this works quite well and makes it easy to see, at a glance, which students are scheduled on what days and times when looking at the calendar. But, typing the student's name as the event title does not automatically add that student as a participant. If you do this, then it will look like that student is scheduled on your calendar when you look at it, but if no specific participants are listed in the event details, then the software considers the event to be a "Global" event. Global events are visible to all students/parents when they log into your site, and they may be confused to see an event on their calendar that is named for another student, so you should not do this unless you are sure it is what you want.
Note, however, that lesson reminders and lesson notes are never sent to anyone for global events. Also, no fees are ever charged for global events, either.
Under normal circumstances, when a student or parent is logged into your studio site, the software is designed to show only events where they (or their child) are listed as a participant. To make sure that this happens, you need to add specific participants to the event when you create it, or add them later, if you don't know for sure who will be participating.
If you create the event as an open event that requires registration, then the event will be visible to all students/parents until the maximum number of students have registered for the event. When the event is full, then it will only show up for students who are listed as participants.
Special cases -
Naturally, there are times when a teacher might want to put an event on the calendar that is not meant for any of the students, but is also not meant to be visible as a global event. In these cases, it is strongly recommended that you create a "test" student in your studio and list that student as the participant for the event. This will effectively hide the event from your students, while still giving you the benefit of having the event listed on the teacher's view of the calendar.
Another special case for this is with the Google calendar sync. If you set up the Google calendar sync, be aware that the link that is generated will show the teacher's calendar, which has all events listed. If you share this link with anyone, they will be able to see all events that are on your calendar, and there is not any way to filter the view.