  Using Resonant Inelastic X-ray Scattering, we measure the paramagnon
dispersion and damping of undoped, antiferromagnetic Ca$_2$CuO$_2$Cl$_2$ as
well as doped, superconducting Na$_{x}$Ca$_{2-x}$CuO$_2$Cl$_2$. Our estimation
of the spin-exchange parameter and width of the paramagnon peak at the zone
boundary $X=(0.5,0)$ confirms that no simple relation can be drawn between
these parameters and the critical temperature $T_\mathrm{c}$. Consistently with
other cuprate compounds, we show that upon doping there is a slight softening
at $(0.25,0)$, but not at the zone boundary $X$. In combination with these
measurements we perform calculations of the dynamical spin structure factor of
the one-band Hubbard model using cluster dynamical mean-field theory. The
calculations are in excellent agreement with the experiment in the undoped
case, both in terms of energy position and width. While the increase in width
is also captured upon doping, the dynamical spin structure factor shows a
sizable softening at $X$, which provides insightful information on the
length-scale of the spin fluctuations in doped cuprates.
