By choice or karma, many graduate students in mathematics will eventually have non-teaching careers. Because of the tremendous variety and diversity of non-academic professions, it is almost impossible to paint a general picture. This is perhaps the biggest misconception about a non-academic career.
Communication skills are even more important outside of classrooms.
An industrial researcher interacts with a wide variety of people including engineers, computer scientists, physicists, chemists and business people. The effectiveness of one's work depends on the ability to convey the power and impact of mathematics as well as its beauty and elegance. It is quite possible to explain mathematics in general terms to non-experts. Even a good colloquium talk involves several different levels of depth. Successful communication not only transfers knowledge and insight helpful to others but also brings up good problems, new directions and interesting ideas.