There are two issues with reporting MCMC fits – verifying geometric ergodicity (i.e. that MCMC estimators for expectation values will be reasonably well behaved after only a finite number of iterations) and then quantifying performance (i.e. how accurate those estimators are). The former considers the behavior of the entire fit whereas the latter considers only the behavior for a specific expectation value.
Rhat near 1 for every function whose expectation value is well-defined is a necessary condition for geometric ergodicity, but a relatively weak one especially if you are running only a few chains. When communicating Rhat results then one should quote any function considered for maximal power. When using Hamiltonian Monte Carlo divergences and the energy fraction of missing information (E-FMI) are much more sensitive diagnostics for geometric ergodicity and consider the joint behavior directly and I strongly recommend that those are considered first and foremost.
Once geometric ergodicity has been verified (or lack of geometry ergodicity optimistically rejected…) then one can consider the performance of individual MCMC estimators which is quantified by the effective sample size. Some functions are more sensitive to the autocorrelations in the Markov chain and some are less sensitive so the effective sample size that manifests in their expectation estimates can strongly vary. Consequently conditioned on geometric ergodicity one has to consider and communicate only the effective sample sizes of the expectation values being used in the analysis. Even better, one can just report the MCMC estimator and its standard error estimator when discussing each relevant expectation value.
That said, keep in mind that we estimate the effective sample size and that estimator need not be well-behaved. The N_eff / N check provides some heuristic verification that the autocorrelations in the Markov chain are small enough that we can accurately estimate the effective sample size for each expectation estimator with a reasonable number of iterations (i.e. the defaults in Stan). In other words, one wants to first check for obstructions to geometric ergodicity, then problems in estimating the effective sample sizes, and then finally reporting the relevant effective sample sizes (or the MCMC estimator standard errors).
I typically report Rhats or N_eff / N results with a verification that they’re all within the desired regimes, but if one wants to report more then I highly recommend showing histograms of all of the individual values instead of summarizing with a minimum or maximum.
For a more thorough discussion see https://betanalpha.github.io/assets/case_studies/rstan_workflow.html.