Bayesian : Comparing means of two posterior samples/ Help a Frequentist Out UPDATE  Thanks for the many thoughtful responses and questions! I've made edits here to clarify further. and also respond to each respondent individually.
Original Post
I have two sets of posterior samples of size 5000 each ($\mu_1$, $sd_1$) and ($\mu_2$, $sd_2$).
These are posterior means and posterior standard deviations respectively. The posterior samples are from different prediction models (say one for sales in Walmart and another for sales in JC Penney) , and from each model I have  5000 samples from the posterior distribution of the prediction percentage error in prediction. NOTE The percentage error is based on posterior predictive sampling for a held out set - i.e. it is a fair measure of generalization error.
I can therefore compute mean percentage error, and the standard deviation of the percentage error. I want to compare whether one model is better at prediction than the other - in other words I want to compare whether the means $\mu_1$ and $\mu_2$ are statistically different.
Option 1) In the two sample t-test, in the frequentist world, I would treat these posterior samples as data and convert $sd_1$ and $sd_2$ into standard errors (by dividing each by the square root of sample size i.e. 5000). And run the test.
Option 2) However, my friend believes that the posterior samples represent the posterior distribution. And that $sd_1$ (and $sd_2$ respectively)is the estimate of the population standard deviation. So no need to divide by the square root of sample size.
Further she argues that I should simply take a difference between the two means  (with arbitrary pairings - I don't get this part fully) and plot the distribution. If the high density interval includes 0, then the two are the statistically the same. Basically,  randomly draw 5000 times a percentage error value from each posterior, compute the difference and then a) plot the differences and b) compute the HPD/HDI and check if 0 lies within the 95% HDI/HPD. If it does include 0, then the two models have similar percentage errors. If not the two are different.
I've seen these two links, but somehow the answers are not quite clicking with me.
Calculating posterior of difference given posterior of two means
How should I compare posterior samples of the same parameter from two Bayesian models?
Any references would be most welcome
Question 1) who do you think is right and why?
Question 2) a reference would help a lot.

As a frequentist you have to construct statistics for which you know the distribution of (or at least the asymptotic distribution of) so that you can calculate p-values in order to do inference.
Now what the appropriate statistic will be depends on the model and assumptions, and how correct the inference will be depends on if the data corresponds to the modeling assumptions (or how robust the inference is to specific deviations).
In your case it looks like you have data with potentially unequal variances. The frequentist approach would be to either do a Welch's t-test or do weighted least squares.
As a Bayesian you don't have to worry about existence of asymptotic results for a particular statistic; if you can correctly produce samples from the posterior you can do direct inference on the parameters in your model.
"But I must take the variances into account don't I?!" you may be thinking. Recall that your samples of $\mu_1$ and $\mu_2$ are from the joint distribution which include the standard deviations. As an exercise for such a simple model you can derive the closed form posterior distributions (assuming conjugate priors) and you will see how the posterior draws of the $\mu$s depend on the variances.
So in that sense your colleague is correct looking at the posterior of the difference is sufficient. Although I think its preferable if the samples come from the same MCMC sampling.
