I did a sub-I in both the MICU and SICU. The structure of both of these units is very different and hospital specific.
MICU - mainly medical patients however, some of the patients may have had minor surgery with predominantly medical problems (example being a patient who had a splenectomy for TTP). The MICU can be staffed by a variety of different physicians. Most are pulmonary critical care specialists, however sometimes 'hospitalists' will staff the MICU.
Just in case you didn't know - critical care fellowships have two routes through which they can be obtainted (you can do three years of internal medicine training then then do a pulmonary critical care fellowship or you can do anesthesia training and then do a critical care fellowship). Not to confuse - these are not the same fellowships.
Some institutions have further specific ICUs such as the CCU (cardiac) and NICU (neuro). This just allows for grouping of specific patients together. It is really a variety of people who take care of these patients. Obviously, a cardiologist will be in charge of the CCU.
SICU - only surgical patients. It depends on where you are as to whether its a closed or open unit. Closed unit means the patient is taken care of by the ICU team only, the primary team may still round on the patient and make recommendations as to how the care should proceed however, the final say is by the ICU attending. An open unit means that the primary surgical team 'takes' care of the patient and the ICU team mainly is there to make sure the patient remains stable and to take care of the day to day stuff. Doesn't mean the ICU team doesn't have an important job, just means that the ICU team is more of a consultant. Kind of hard to explain - hopefully this has made some sense.
The SICU can be run by a whole slew of different doctors - mainly it is surgeons or anesthesiologists. In our ICU the attendings are mainly surgeons with a combination of trauma surgeons and general surgeons.
If you are interested in critical care you should try and do an elective in the ICU. I really enjoyed both ICUs however in the MICU I was given a lot more freedom - because the patient was 'mine' (meaning I made all the decisions - with my senior residents help of course). In the SICU it was a little different because all of the patients were the whole teams (ie I wasn't always taking care of the same patient). What this boiled down to was more autonomy in the MICU. This could be completely institution dependent.
Also, I learned a lot more about vents and pathophysiology in the MICU b/c the pulmonary critical care attendings really drilled us with the vent stuff (which is still a little fuzzy for me). In the SICU I learned a lot more about surgical problems and post op care.
Hope this helps