Standard present-day large-scale structure (LSS) analyses make a major assumption in their Bayesian parameter inference - that the likelihood has a Gaussian form. For summary statistics currently used in LSS, this assumption, even if the underlying density field is Gaussian, cannot be correct in detail. We investigate the impact of this assumption on two recent LSS analyses: the Beutler et al. (2017) power spectrum multipole (Pℓ) analysis and the Sinha et al. (2017) group multiplicity function (ζ) analysis. Using non-parametric divergence estimators on mock catalogs originally constructed for covariance matrix estimation, we identify significant non-Gaussianity in both the Pℓ and ζ likelihoods. We then use Gaussian mixture density estimation and Independent Component Analysis on the same mocks to construct likelihood estimates that approximate the true likelihood better than the Gaussian pseudo-likelihood. Using these likelihood estimates, we accurately estimate the true posterior probability distribution of the Beutler et al. (2017) and Sinha et al. (2017) parameters. Likelihood non-Gaussianity shifts the fσ8 constraint by −0.44σ, but otherwise, does not significantly impact the overall parameter constraints of Beutler et al. (2017). For the ζ analysis, using the pseudo-likelihood significantly underestimates the uncertainties and biases the constraints of Sinha et al. (2017) halo occupation parameters. For logM1 and α, the posteriors are shifted by +0.43σ and −0.51σ and broadened by 42% and 66%, respectively. The divergence and likelihood estimation methods we present provide a straightforward framework for quantifying the impact of likelihood non-Gaussianity and deriving more accurate parameter constraints.