First, let me tell you that I went to neither Mt. Sinai nor Duke for med school, but I've been a resident at each place, and dealt with med students from each.
Comparatively, the structure is different between the schools as far as curriculum, as Duke is preclinical/clinical/research/clinical - which means you're on surgery as an MSII, after having had 8 weeks (only) of anatomy. Students DO complete their research in surgery (either at Duke or elsewhere), so they make more connections there. As an MS4, it's gravy - as far as I know, last year they put in a requirement that you had to do a month of some type of critical care - either MICU, anesthesia, or EM. 4th year students get assigned to a team or a surgeon (I'm not sure how specific this goes - I've seen MS4's on purple - transplant, that are right in the guts of a kidney or liver transplant, involved beginning to end, and others assigned to a specific surgeon that operates 2 days a week, and the students are ghosts), and are true sub-I's.
Mt. Sinai is the preclinical/preclinical/clinical/clinical model. Your main time is at the Mt. Sinai Hospital and the largest affiliate, Elmhurst. I never dealt with a surgical attending at Elmhurst, since I was IM-prelim going into EM, so I can't speak as much about the attendings and structure.
Both schools put people into good surgery spots, but, head-to-head, I think Duke has more juice (since I've seen weaker students get good spots from Duke). One thing I've seen only anecdotally (and may have no relevance) that was told to me by several residents at MSH is the "Jewish pipeline" - it's probably selection bias on the part of the applicants, but there may also be some preference for Jewish applicants (especially observant ones).