  Optical spectroscopy, X-ray diffraction measurements, density functional
theory (DFT) and density functional theory + embedded dynamical mean field
theory (DFT+eDMFT) have been used to characterize structural and electronic
properties of hexagonal $M_2$Mo$_3$O$_8$ ($M$=Fe, Mn) polar magnets. Our
experimental data are consistent with the room temperature structure belonging
to the space group P6$_3$mc for both compounds. The experimental structural and
electronic properties at room temperature are well reproduced within DFT+eDMFT
method, thus establishing its predictive power in the paramagnetic phase. With
decreasing temperature, both compounds undergo a magnetic phase transition and
we argue that this transition is concurrent with a structural phase transition
(symmetry change from P6$_3$mc) in the Fe compound and an isostructural
transition (no symmetry change from P6$_3$mc) in the Mn compound. In addition,
the unusual temperature dependent behavior of electronic d-d transitions in
Fe$^{2+}$ ions is discussed.
