All 280 of the statistically-complete Palomar sample of nearby (<120 Mpc) galaxies δ > 20○ have been observed at 1.5 GHz as part of the LeMMINGs e-MERLIN legacy survey. Here, we present Chandra X-ray observations of the nuclei of 213 of these galaxies, including a statistically-complete sub-set of 113 galaxies in the declination range 40○ <δ < 65○. We observed galaxies of all optical spectral types, including ‘active’ galaxies (e.g. LINERs and Seyferts) and ‘inactive’ galaxies like HII galaxies and absorption line galaxies (ALG). The X-ray flux limit of our survey is 1.65 × 10−14 erg s−1 cm−2 (0.3−10 keV). We detect X-ray emission coincident within 2-arcsec of the nucleus in 150/213 galaxies, including 13/14 Seyferts, 68/77 LINERs, 13/22 ALGs and 56/100 HII galaxies, but cannot completely rule out contamination from non-AGN processes in sources with nuclear luminosities ≲1039 erg s−1. We construct an X-ray Luminosity function (XLF) and find that the local galaxy XLF, when including all AGN types, can be represented as a single power-law of slope −0.54 ± 0.06. The Eddington ratio of the Seyferts is usually 2−4 decades higher than that of the LINERs, ALGs and HII galaxies, which are mostly detected with Eddington ratios ≲10−3. Using [OIII] line measurements and BH masses from the literature, we show that LINERs, HII galaxies and ALGs follow similar correlations to low luminosities, suggesting that some ‘inactive’ galaxies may harbour AGN.