From the viewpoint of an academic faculty head, UK:

The student needs to put their grievance in writing. This will:
a) formalise the paper trail of evidence for the grievance,
b) crystallise the key point of the issue for the student, and
c) show that they are serious in pursuing the grievance.

Once passed to me, the letter remains confidential between the two of us.

I would then interview the student and investigate the matter, making no promises:

I observe every member of staff and every student cohort once per term; I would ensure that I observed that member of staff with that student group at the earliest possible opportunity as part of the routine programme.

I take one whole-class tutorial with every cohort per term. I would ensure that also occurred as rapidly as practicable and I would raise a general question on staff conduct, and see what response I get, both from the students and the particular individual.

Based on the evidence gathered:

a) the student may be a sensitive soul and they alone have had their sensibilities offended - speak to the member of staff, using my observation as my reason, and invite them to temper their manner.

b) the member of staff is attempting an approach which is not appropriate for a significant minority of the class, but which is nonetheless not an incorrect or unprofessional one - speak to them, using both the observation and tutorial as evidence, and formalise my "advice" to them.

c) the member of staff is being unprofessional and/or offensive. Take immediate disciplinary action. This is the only circumstance where the student's letter might enter the public domain, and then only late on in disciplinary proceedings.

d) no evidence to suggest any conduct problem whatsoever with member of staff (i.e. not even anything which might offend a sensitive soul) - explain to student that I can find no evidence to support their complaint. They have the option of approaching the Vice Principal at this stage.

e) evidence of a malicious prosecution of the member of staff - discuss with the student in the strongest terms and advise the member of staff, naming no names.

If the student were to come to me with a tape recording made without staff permission, they would be subject to disciplinary action themselves.

Please note that the nuts and bolts of the process would not be explained to the student in advance, I would want to collect unsolicited opinions and not those of students who may have been primed or influenced to give certain views.

I cannot comment on the process across the pond, only that given the litigious culture, I cannot imagine it would be much different.