How does Stan choose priors for standard deviation parameters? Are they treated differently from other kinds of parameters? Any documentation of how Stan chooses priors will be appreciated.

Stan doesnâ€™t choose priors for parameters. However, any parameters that you donâ€™t assign a prior to, are implicitly given a uniform prior

And that â€śuniformâ€ť prior is usually improper.

Yes! and there goes my question about what kind of prior is given to a standard deviation parameter. Is there any special treatment for std deviation parameters? I learnt that all parameters that are not given any bounded support or a prior are assigned an improper uniform prior, but for std deviation parameters, STAN samples from their log, with a jacobian adjustment for the transformation. How true is this statement? Any references?

See this section of the manual for more background on the lower bound constraint and sampling: 10.2 Lower bounded scalar | Stan Reference Manual

I got new different knowledge from that manual but it still has not addressed my question about how STAN chooses priors for standard deviation parameters when no bounds or priors have been assigned. Does Stan assign the same kind of uniform improper priors to such parameters or is there some other special treatment under the hood because of the nature of standard deviation parameters.

There goes my question about what kind of prior is given to a standard deviation parameter. Is there any special treatment for std deviation parameters? I learned that all parameters that are not given any bounded support or a prior are assigned an improper uniform prior, but for std deviation parameters, STAN samples from their log, with a jacobian adjustment for the transformation. How true is this statement? Any references?

There is no special treatment for any parameter in stan. When you donâ€™t specify a prior it is uniform. If you specify a parameter with a lower bound of zero, then the prior is uniform from 0 to positive infinity.

For parameters specified with a lower bound, this is achieved using the log transformation, as discussed in my previous link