I'm not sure about other lines, but I worked as a domestic violence counselor at one point during school. Answering the crisis line was one of the responsibilities of the job. That particular line did not take volunteers but relied on employees and interns to man the lines. We not only had specific DV Crisis training, but were all also in school for some aspect of mental health work or another. We processed calls in supervision.

I can imagine many crisis lines offer similar debriefing to their volunteers. It's not so much giving specific details as it is going over the volunteer's/employee's responses, and any emotions that may be brought up by responding to a specific call... I know most lines honor confidentiality within the bounds of the laws of that state/region. Most do not record calls or track the calls outside of keeping data on the number received and the nature of the calls. They know a lot of people would not call if their information was being recorded and kept. The DV crisis line I worked for recorded info, but only if the callers chose to give it. I had a form I needed to fill out, but I had no way of knowing the accuracy of the info being given me. We did that because it was often the way clients entered our services (and we had to report our stats to the national domestic violence services providers organization (I think it's DDVAC?) for accreditation purposes).

I rarely give accurate personal info when calling a crisis line unless it's connected to a specific provider of mine...