  We present an improved study of the relation between supermassive black hole
growth and their host galaxy properties in the local Universe (z < 0.33). To
this end, we build an extensive sample combining spectroscopic measurements of
star-formation rate (SFR) and stellar mass from Sloan Digital Sky Survey, with
specific Black Hole accretion rate (sBHAR, $\lambda_{\mathrm{sBHAR}} \propto
L_{\mathrm{X}}/\mathcal{M}_{\ast}$) derived from the XMM-Newton Serendipitous
Source Catalogue (3XMM-DR8) and the Chandra Source Catalogue (CSC 2.0). We find
that the sBHAR probability distribution for both star-forming and quiescent
galaxies has a power-law shape peaking at $\log\lambda_{\mathrm{sBHAR}}\sim
-3.5$ and declining toward lower sBHAR in all stellar mass ranges. This finding
confirms the decrease of AGN activity in the local Universe compared to higher
redshifts. We observe a significant correlation between
$\log\,\lambda_{\mathrm{sBHAR}}$ and $\log\,{\mathrm{SFR}}$ in almost all
stellar mass ranges, but the relation is shallower compared to higher
redshifts, indicating a reduced availability of accreting material in the local
Universe. At the same time, the BHAR-to-SFR ratio for star-forming galaxies
strongly correlates with stellar mass, supporting the scenario where both AGN
activity and stellar formation primarily depend on the stellar mass via
fuelling by a common gas reservoir. Conversely, this ratio remains constant for
quiescent galaxies, possibly indicating the existence of the different physical
mechanisms responsible for AGN fuelling or different accretion mode in
quiescent galaxies.
