  By making use of a covariant formulation of the chiral kinetic theory in the
relaxation-time approximation, we derive the first-order dissipative
hydrodynamics equations for a charged chiral plasma with background
electromagnetic fields. We identify the global equilibrium state for a rotating
chiral plasma confined to a cylindrical region with realistic boundary
conditions. Then, by using linearized hydrodynamic equations, supplemented by
the Maxwell equations, we study hydrodynamic modes of magnetized rotating
chiral plasma in the regimes of high temperature and high density. We find
that, in both regimes, dynamical electromagnetism has profound effects on the
spectrum of propagating modes. In particular, there are only the sound and
Alfv\'en waves in the regime of high temperature, and the plasmons and helicons
at high density. We also show that the chiral magnetic wave is universally
overdamped because of high electrical conductivity in plasma that causes an
efficient screening of charge fluctuations. The physics implications of the
main results are briefly discussed.
