OK - Thanks for your help.
I have thought more about this, but still I can’t come up with a good answer. In normal model case
\log(w_i)=-\log(p(y_i|\mu,\sigma)) has gamma distribution if \mu has normal distribution and \sigma is fixed. The distribution can be almost gamma also if \sigma is not fixed based on the asymptotic results by Watanabe, but that would imply that it has close to gamma distribution mostly in easy cases where waic and psis-loo work also well. Sorry, not being able to help more.
Relevance of the standard error in the "loo" package