  The gravitational waves (GWs) emitted by inspiraling binary black holes,
expected to be detected by the Laser Interferometer Space Antenna (LISA), could
be used to determine the luminosity distance to these sources with the
unprecedented precision of <~ 1%. We study cosmological parameter constraints
from such standard sirens, in the presence of gravitational lensing by
large-scale structure. Lensing introduces magnification with a probability
distribution function (PDF) whose shape is highly skewed and depends on
cosmological parameters. We use Monte-Carlo simulations to generate mock
samples of standard sirens, including a small intrinsic scatter, as well as the
additional, larger scatter from lensing, in their inferred distances. We derive
constraints on cosmological parameters, by simultaneously fitting the mean and
the distribution of the residuals on the distance vs redshift (d_L - z) Hubble
diagram. We find that for standard sirens at redshift z ~ 1, the sensitivity to
a single cosmological parameter, such as the matter density Omega_m, or the
dark energy equation of state w, is ~ 50%-80% tighter when the skewed lensing
PDF is used, compared to the sensitivity derived from a Gaussian PDF with the
same variance. When these two parameters are constrained simultaneously, the
skewness yields a further enhanced improvement (by ~ 120%), owing to the
correlation between the parameters. The sensitivity to the amplitude of the
matter power spectrum, sigma_8 from the cosmological dependence of the PDF
alone, however, is ~ 20% worse than that from the Gaussian PDF. At higher
redshifts, the PDF resembles a Gaussian more closely, and the effects of the
skewness become less prominent. These results highlight the importance of
obtaining an accurate and reliable PDF of the lensing convergence, in order to
realize the full potential of standard sirens as cosmological probes.
