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)

You can use the following trick to swap stdout and stderr. Then you just use the regular pipe functionality.

( proc1 3>&1 1>&2- 2>&3- ) | proc2

Provided stdout and stderr both pointed to the same place at the start, this will give you what you need.

What the x>&y bit does is to change file handle x so it now sends its data to wherever file handle y currently points. For our specific case:

• 3>&1 creates a new handle 3 which will output to the current handle 1 (original stdout), just to save it somewhere for the final bullet point below.
• 1>&2 modifies handle 1 (stdout) to output to the current handle 2 (original stderr).
• 2>&3- modifies handle 2 (stderr) to output to the current handle 3 (original stdout) then closes handle 3 (via the - at the end).

It's effectively the swap command you see in sorting algorithms:

temp   = value1;
value1 = value2;
value2 = temp;