They need not be mathematicians, but they should take pleasure in beautiful proofs and demonstrations, and in showing these to their students. They should be the kind of people who like to read popularized mathematics books, such as the kind the late Martin Gardner wrote, or that Ian Stewart or Keith Devlin write.
Of course, this is a counsel of perfection. We have to start where we are, and try to ensure, for the moment, that every child is exposed to at least one teacher like this in his or her first years of school, and that there is then some institutionalized path that children who become interested in mathematics can follow: Maths Clubs, for instance.
Of course, good teachers will know, and use, whatever mathematical abilities they discover latent in their students. But mathematics has to be taught. The idea that it is just lying, latent, in our pupils, is wrong (despite Socrates' famously, supposedly, demonstrating this the Meno).