We present $Chandra$ X-ray point source catalogs for 9 Hickson Compact Groups (HCGs, 37 galaxies) at distances $34 - 89$ Mpc. We perform detailed X-ray point source detection and photometry, and interpret the point source population by means of simulated hardness ratios. We thus estimate X-ray luminosities ($L_X$) for all sources, most of which are too weak for reliable spectral fitting. For all sources, we provide catalogs with counts, count rates, power-law indices ($Gamma$), hardness ratios, and $L_X$, in the full ($0.5-8.0$ keV), soft ($0.5-2.0$ keV) and hard ($2.0-8.0$ keV) bands. We use optical emission-line ratios from the literature to re-classify 24 galaxies as star-forming, accreting onto a supermassive black hole (AGNs), transition objects, or low-ionization nuclear emission regions (LINERs). Two-thirds of our galaxies have nuclear X-ray sources with $Swift$/UVOT counterparts. Two nuclei have $L_{X,{rm 0.5-8.0 keV}}$~$ > 10^{42}$ erg s$^{-1}$, are strong multi-wavelength AGNs and follow the known $alpha_{rm OX}- u L_{ u,rm near UV}$ correlation for strong AGNs. Otherwise, most nuclei are X-ray faint, consistent with either a low-luminosity AGN or a nuclear X-ray binary population, and fall in the non-AGN locus in $alpha_{rm OX}- u L_{ u,rm near UV}$ space, which also hosts other, normal, galaxies. Our results suggest that HCG X-ray nuclei in high specific star formation rate spiral galaxies are likely dominated by star formation, while those with low specific star formation rates in earlier types likely harbor a weak AGN. The AGN fraction in HCG galaxies with $M_R le -20$ and $L_{X,{rm 0.5-8.0 keV}} ge 10^{41}$ erg s$^{-1}$ is $0.08^{+0.35}_{-0.01}$, somewhat higher than the $sim 5%$ fraction in galaxy clusters.
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