I've seen externship programs where doctoral students are hired for staff positions (e.g., psychometrist) and told "what your training program thinks of the time you spend here is none of our business." I always had an arm's length relationship with these programs and can't comment on their legality or ethics.
When I co-treat or co-lead a visit with an extern, I bill for that time. I also bill for the time I spend either editing or rewriting trainee-written clinical reports. I don't bill for the trainee's time spent either writing the first draft or implementing my edits.
On our inpatient unit, my trainees each carry a small caseload. When first meeting a patient, they typically complete an interview and cognitive/emotional screen at bedside, as well as a collateral interview (if appropriate/feasible)--externs then send me their notes/a draft note based on that initial visit. I then meet with all patients seen by my supervisees during my own walking rounds and complete my own, independent brief screen (e.g., orientation, mood) and brief interview. I typically bill for these visits using a non-time-based code (e.g., 96156), incorporating information provided by the trainee. This is a scenario where partnering with trainees does increase my efficiency, while simultaneously allowing trainees to obtain supervised experience in a unique/acute medical setting (with the added bonus of giving them the opportunity to generate many integrated reports--typically at least one-to-two per day).