NOT A TEACHER
(1) I believe that here in the United States, most people say:
(a) Walk on the street. = Walk on the SIDEWALK.
(i) Pardon this crude example, but it might help you remember: sometimes prostitutes are called streetwalkers.
(b) Walk in the street. = You are literally (actually) walking in the area used by automobiles. Mothers will often warn their children: Don't play in the street!