There are many types of independent schools and qualifications for teaching candidates vary considerably. At a minimum, most schools are looking for candidates with a bachelor’s degree, ability to teach at several levels within one subject area, ability to work with students of different capabilities, and enthusiasm for your area of concentration. Prior teaching experience is highly desirable but not required. Schools also value a candidate's experience supervising children in camp, in a childcare setting, coaching sports, or tutoring. State certification is usually not required.  

Candidates tend to increase their desirability when they can articulate and demonstrate an interest in working with young people—in the classroom, as advisors, and in co-curricular activities and athletics. Schools are looking for people with good communication skills, who are curious and flexible, and who want to improve and grow professionally. Being offered the right job depends on being able to demonstrate to a school that your talents and interests are the right match for its philosophy and needs.

From "About Independent Schools -- An Overview" by Liz Fernandez, Diversity Job Fair Administrator,
New York State Association of Independent Schools (NYSAIS)