There are also questions tagged rstan and pystan and brms, etc. on StackOverflow. We have fewer questions asked on StackOverflow than we would if we didn’t have the dedicated Stan forums here.
From July 2015 to this week, RStan for @mcmc_stan has been downloaded from RStudio CRAN mirror one million times (loo 630K times). Python users love much more downloading as PyStan gets 700K+ downloads just in the last month.
We don’t know how many @mcmc_stan users these imply, but during the last month RStan downloads were 6% of ggplot2 downloads (200% of R tensorflow) and PyStan downloads were 1.6% of numpy downloads (9% of matplotlib and 14% of tensorflow downloads).
In 2019 it was estimated that there are 8 million Python developers (https://www.zdnet.com/article/programming-languages-python-developers-now-outnumber-java-ones/). If half of them use numpy, then we get an estimate of 64K PyStan users.
In 2014 it was estimated that there are 2+ million R users (https://blog.revolutionanalytics.com/2014/04/a-world-map-of-r-user-activity.html). Assuming conservatively the same number in 2019 and half of them download ggplot2, then we get estimate of 60K RStan users.
Combined estimate would be more than 120K @mcmc_stan users plus CmdStan, StatStan, JuliaStan and MatlabStan users. Of course it is possible that Stan users download much more often than average users which would inflate these estimates.
Also it’s difficult to estimate how many users use pre-installed Stan in cloud services or how many “users” are students who are forced to use Stan in a course (e.g. https://github.com/avehtari/BDA_course_Aalto/) and then never use it after that. But the order of magnitude of the above estimates is likely to be correct.
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But that’s just the RStudio mirror. I don’t use it as I don’t use RStudio.
I’ve been telling people “roughly 100K”. There are about 4K Google Scholar citations now I can find and about 3K users registered in the forums.