What information can Soar Scribe access and view from the calendars I link? I'm concerned about preserving the privacy of meetings without video/web calls.

When you connect your calendar, it will tell you what it is requesting permission for. It requests read only permission to all calendar events. It has to scan the event to look for meeting links to record so it does look at all your calendar events.  That's how it knows which ones have recordable meeting links.