The CBC/HB software outputs standard deviations of the draws for respondents for each utility level in the model (but only if you do not request "draws" as an advanced option). That's the equivalent precision measure in the Bayesian sense at the individual level.
For the standard deviation of the population means (the standard error from the frequentist world), the software doesn't provide this automatically, but you can compute it pretty easily from the output. Open the _alpha.csv file that contains the population mean estimates at each iteration (the estimate of alpha at each iteration). Ignore the first few thousand iterations prior to convergence. Then, compute the standard deviation of the alpha draws after convergence for each attribute level. This gives you the standard deviation of the population mean estimates for each parameter.