No, if both persons are in inertial frames of reference the
situation is completely symmetric so the 'paradox' does not occur.
Also note that it is not really a paradox because general
relativity has a conclusive answer to what happens to the
twins.
It is; however, not possible to set up a twin paradox-like
situation with neither twin never leaving an inertial frame. This
is because if they want to move apart, and come back again, they
need to accelerate somehow, and the act of acceleration causes you
to leave an inertial frame.