This paper presents an analytic formulation of the near- and far-field distributions due to periodic spherical arrays of dipole antennas on two spherical biisotropic substrates, i.e., on the chiral sphere and on a perfectly conducting sphere coated with one-layer chiral medium, respectively. For an arbitrary orientation of the dipole antenna array, the dyadic Green's functions in the form of an eigenfunction expansion in terms of the normalized spherical vector wave functions are directly employed. Thereby, exact representations of the radiated fields in both near and far zones are obtained in closed form. They are all in a superposition of right- and left-handed circularly polarized waves, and are also applicable for the cases of spherical arrays of dipole antennas on spherical isotropic substrates. Furthermore, the constitutive parameter effects of the chiral medium on the three-dimensional patterns of the far-zone field and its polarized feature are demonstrated numerically.