Actually, all the hospitals have trainees, but I guess they are at a certain level before they are allowed to do anything surgical. Nevertheless, I have to say that my experience was that the experienced doctors at United Christian have better bedside manner, which is weird...you'd think the experienced ones would be impatient and arrogant but it's the other way around.
Are you having a c-sec? If so, you'd have a trainee doing some part of it but in the presence of a senior.
If a standard vaginal delivery, mainly the midwives do everything including stitching you up. They're pretty good.
That said, I think the docs in private can do smaller incisions and neater stitches - having compared my c-sec to episiotomy to others - but that's the difference between paying 60,000 -100,000 vs free, I guess.