I've not had the privilege of working with Anne Bogart, but I did see her at a master class, and I've worked with other people who have branched out from the SITI company. They're all idealists in the best sense of the world.
I have worked with several directors that I had been warned about beforehand; the producer who hired me said "He's a real bear" or I hear from a friend that "she's impossible to work with."
In all these cases, my personal experience with these directors was that they expected excellence; they got their reputation by impatience with mediocrity. When I did my job to the best of my ability, and responded to their requests in a timely manner, we got along famously. I also made an effort to cover the director's back with the other areas; if the director wasn't happy with props, I'd spend extra time the next day building my relationship with the propsmaster and articulating in an actionable and respectful way the notes that needed to be done; notes that some directors lack the tact to communicate. When I saw a problem on the horizon, I'd try to give a heads-up to the department it concerned so that they didn't feel broadsided by a director blow-up. When the director started second-guessing a design choice that would have meant a huge re-design at the eleventh hour, I was part of a group that re-affirmed the original design choice, because it was frickin' awesome, not because changing it would have meant a lot of work.
I worked hard at understanding what the director wanted, even when their way of working was alien to me. I went through the thought process of an asst. director - even though I never did anything that an AD would do, it helped me to foresee what problems would arise. My motto is always "Give them what they need AS WELL AS what they ask for."
I was scared, sure - but I never let it show. Directors and actors maybe aren't prevented from bringing their emotional baggage to rehearsal, but the SM has to be even-tempered, especially if they're being attacked or are under pressure.
I prepared as much as I could. For one of these shows, I even took my prompt book home and practiced a tough cueing sequence BEFORE we went into tech. I've never done that before, or since. But it saved my bacon during that tech...
So, to sum up, Scary Directors - respond with preparation, relationship-building, confidence, and above all, calmness.