We assume independence in the choice tasks for our pooled logit. We don't cluster like you mention.
For HB, because most users of our software (practitioners in industry) are accustomed to Frequentist statistics, we take the simple route of computing the standard deviation across the posterior means of the draws for each respondent. In other words, we create a single point estimate vector of betas for each respondent. This is NOT the Bayesian way of doing things, for sure. It probably understates the true standard errors.
However, our HB software does produce a variance-covariance matrix for the alphas (the population means estimates), which would be more appropriate to examine for Bayesians. Better yet, you can run histograms and compute deciles on the draws of alpha (after convergence has been assumed) that are made available to you in an alpha draws file by our software.