From the kid they'll want to talk about their interests, strengths/weaknesses, stuff you discussed in your statement about the kid, they'll want to know why the kid is interested in the school, why you think the kid would be a good fit, what you like about the school, what your questions are. We're applying to private MS too and have only done one interview so far and have three more - the first was just with the kid - it was more like a conversation, tell me about this, tell me about that... For us I think they'll likely want to know the same - and get a feel for what kind of people we are and if it would be a good fit.
From the kid they'll want to talk about their interests, strengths/weaknesses, stuff you discussed in your statement about the kid, they'll want to know why the kid is interested in the school, why you think the kid would be a good fit, what you like about the school, what your questions are. We're applying to private MS too and have only done one interview so far and have three more - the first was just with the kid - it was more like a conversation, tell me about this, tell me about that... For us I think they'll likely want to know the same - and get a feel for what kind of people we are and if it would be a good fit.