  We aim to characterize the nature of galaxies whose optical emission line
diagnostics are consistent with star formation, but whose X-ray properties
strongly point towards the presence of an AGN. Understanding these sources is
of particular importance in assessing the completeness of AGN samples derived
from large galaxy surveys, selected solely on the basis of their optical
spectral properties.We construct a large sample of 211 NELGs, which have FWHMs
Hb emission line <1200 km/s from the SDSS-DR7 galaxy spectroscopic catalogue,
for which we are able to construct a classical diagnostic diagram, [OIII]/Hb
versus [NII]/Ha (hence z<0.4), and that are also detected in the hard energy
band and present in the 2XMM catalogue. This sample offers a large database by
which to investigate potential mismatches between optical diagnostics and X-ray
emission. Among these 211 objects, which based on our selection criteria are
all at z<0.4, we find that 145 galaxies are diagnosed as AGNs, having 2-10 keV
X-ray luminosities that span a wide range, from 10^40 erg/s to above 10^44
erg/s. Out of the remaining 66 galaxies, which are instead diagnosed as SF, we
find a bimodal distribution in which 28 have X-ray luminosities in excess of
10^42 erg/s, large T (>1), and large X/O ratio (>0.1), while the rest are
consistent with being simply SF galaxies. Those 28 galaxies exhibit the
broadest Hb line FWHMs, from ~300 to 1200 km/s, and their X-ray spectrum is
steeper than average and often displays a soft excess. We therefore conclude
that the population of X-ray luminous NELGs with optical lines consistent with
those of a starforming galaxy (which represent 19% of our whole sample) is
largely dominated by NLS1s. The occurrence of such sources in the overall
optically selected sample is small (<2%), hence the contamination of optically
selected galaxies by NLS1s is very small.
