As already mentioned, this is going to vary based on the specific hospital. Typically, NP's who are hired in by a surgical subspecialty will be asked to function within the spectrum of settings the surgeon sees his or her patients - that can include seeing preop patients in the clinic, assisting in the surgery itself (for some), following up on the patient in the PACU or SICU, all the way to transfer to floor, discharge, and clinic follow-up. That would be an ideal role but again there would variations in that pattern. You may be referred to as the NP for General Surgery or Colorectal Surgery or Hepato-biliary Surgery, etc depending on the subspecialty you work under.
SICU's are not all the same. In Trauma Centers, you may see a great deal of traumas on top of the bread and butter ACS (Acute Care Surgery) patients who had abdominal surgeries. In hospitals with a high cancer population, you will see Gyn-Onc, Surg-Onc, ENT cases in the SICU. Transplant centers have post solid organ transplants in their SICU's (mainly livers). The SICU is typically the purview of the intensivist and in many places that intensivist is not always a surgeon (in ours, they are a combo of surgeons and anesthesiologists with Critical Care training). The NP's who work with these physicians only work in the SICU and are mostly referred to as Critical Care NP's.