  We present a detailed study of the optical spectroscopic properties of 12
active galactic nuclei (AGNs) with candidate low-mass black holes (BHs)
selected by Kamizasa et al. through rapid X-ray variability. The high-quality,
echellette Magellan spectra reveal broad H$\alpha$ emission in all the sources,
allowing us to estimate robust viral BH masses and Eddington ratios for this
unique sample. We confirm that the sample contains low-mass BHs accreting at
high rates: the median $M_{\rm BH} = 1.2\times 10^6M_\odot$ and median $L_{\rm
bol}/L_{\rm Edd}=0.44$. The sample follows the $M_{\rm BH}-\sigma_*$ relation,
within the considerable scatter typical of pseudobulges, the probable hosts of
these low-mass AGNs. Various lines of evidence suggest that ongoing star
formation is prevalent in these systems. We propose a new strategy to estimate
star formation rates in AGNs hosted by low-mass, low-metallicity galaxies,
based on modification of an existing method using the strength of [O II]
$\lambda 3727$, [O III] $\lambda 5007$, and X-rays.
