This is  g(s_1,s_2,..s_m) = (g_1(s_1,..,s_m),...,g_N(s_1,...,s_m). Sorry!

This conjecture is false: consider for example the functions

f(t_1,t_2) = (cos(t_1) cos(t_2), sin(t_1) cos(t_2)) from R^2 to R^2 and

g(s_1,s_2) =  (1+ cos(s_1) cos(s_2), sin(s_1) cos(s_2)) from R^2 to R^2.

The image of f is the disk of radius 1 and center (0,0) in the plane  while the image

of g is the disk of radius 1 and center (1,0).

The set of (t_1,t_2) such that there exists (s_1,s_2) s.t. f(t_1,t_2)=g(s_1,s_2) is not R^2 and has not an empty interior set.

This closes the question.