Scrub nurses are part of the sterile field, and therefore cannot administer blood products without breaking sterility. The way things work at my facility, anesthesia and the circulator check the blood unit, but anesthesia is responsible for starting it.
The way you need to think about it is how the surgical site is sterile, the surgeon is sterile, and the scrub person (may be an RN, may not be an RN- there are surgical technologist programs that graduate people able to take this role as well as some old-timers who had on-the-job training to scrub) is sterile. They only work with sterile items (instruments, sponges, sutures). The circulator is unsterile, does not touch any part of the sterile field, and may work with such things as IVs, equipment in the room, and any other tasks not requiring him/her to be sterile- with the exception of foley insertion and prepping, which only require sterile gloves.
P.S. Nurses who scrub are OR nurses. Many places will not distinguish between the roles when hiring. It's not job as just circulator or job as just scrub. If you are an RN, you are expected, as a minimum, to circulate. If you are able to learn to scrub, you can fulfill that role as well. However, AORN standards, CMS requirements, and several states require an RN circulator. Scrub personnel are not required to be RNs. Many hospitals hire surgical technologists to fulfill the scrub role, who generally get paid less, and hire fewer nurses, requiring them to primarily fill the role of circulator. Right now my facility is about a 40% RN staff and 60% ST staff. RNs very rarely scrub.
I think you really need to spend some time shadowing in the OR to understand the role(s) of the RN in the OR.