With bash, how can I pipe standard error into another process?

There is also process substitution. Which makes a process substitute for a file.
You can send stderr to a file as follows:

process1 2> file

But you can substitute a process for the file as follows:

process1 2> >(process2)

Here is a concrete example that sends stderr to both the screen and appends to a logfile

sh myscript 2> >(tee -a errlog)