  We present a comparative analysis of 1$s$ and 2$p$ core-level hard X-ray
photoelectron spectroscopy (HAXPES) spectra in metallic VO$_2$ and CrO$_2$.
Even though the V 1$s$ and 2$p$ spectra in VO$_2$ display similar line shapes
except the absence or presence of a spin-orbit coupling splitting, the Cr 1$s$
and 2$p$ spectra exhibit distinct main-line shapes. The experimental HAXPES
spectra are analyzed by the Anderson impurity model based on the density
functional theory + dynamical mean-field theory and a conventional MO$_6$
cluster model. We elucidate the complex interplay between formation of the
intra-atomic multiplet and charge transfer effect on the chemical bonding
followed by the 1$s$ and 2$p$ core electron excitations. We demonstrate the
advantage of the 1$s$ excitations to the routinely-employed 2$p$ excitations
for distinguishing between metal-ligand and metal-metal charge transfer
contributions in early 3$d$ transition-metal oxides.
