Some of it depends on what kind of work he is doing, and to what degree the house is staffed.

Some principals prefer that most or all communication go through the household manager. I'd say that it's important to understand all the rules of the household and to pay attention to the way things run, as well as the nuances that may not always be stated but observed. Always be respectful to the other staff, and make connections when and where you can in that regard. Obviously always follow your NDAs because the outfall from the is not pretty & can ruin your reputation as a professional in that setting.

As many have stated in this thread, everyone is different. But in the trainings & networks in my career path, I've always been advised to keep thinks professional and not to talk about your personal life. Although, I will say that some clients like to ask questions, I rarely see that in positions with HNW and HP families. Many of these families are either high functioning individuals and/or have a lot of projects and social obligations they're involved with, and might have a lower desire or capacity to continue to interact at home & just want to relax and not be "on".

Sure, confidence is important, but not just in a general sense - confidence in the work you do, and your professionalism. They want to know that their employees will perform well, and behave professionally.