SY_Mom,
I don't have the experience for private school interview yet since my 2 little ones are not there yet. But, I have a coworker who has applied for 7 private schools last year for her daughter and she's so "generous" to share her experience to me "everyday". Here're some ideas and hope that's useful to you (BTW, all the schools she applied are in SF).
1) Almost every school required her daughter to go for 1-2 playdays, 1-to-1 with the teacher(s) and/or a group play with other kids. The teachers sort of observed how her daughter response to instruction and how well she can socialize with the others.
2) For parent interviews, there're so many different formats; a panels including principals and teachers; parents and principals only; group discussions with other parents...My coworker had one distinctive interview with a school principal - the principal greeted her and asked her what my coworker wanted to say or ask. My coworker just opened her mouth and stucked there for at least 10 seconds.
3) Some schools asked my coworker for an essay writing about her visions and plans for her daughter's education.
4) Information about the kid's and family activities should be shared if suitable such as photos taken during family trips (my coworker goes for at least 2 family trips every year), drawings of the kid, etc.
I cannot remember all the details as there're just too much for me to hear what she said. But, I believe that every school is different. Maybe you can take a look of the school's mission statements and you can find some hints. Good luck.
花旗太太生活在花旗國,留意時差,你問我未必即時答到。