At least in my state, as a former emergency services worker, our EVIP (emergency vehicle incident prevention) course (which serves as a "substitute" for requiring a CDL to drive an emergency vehicle) absolutely tells you that driver liability can and will flow through to your personally, even when driving a work vehicle for work purposes. (Because that also comes with a presumption of liability for any incident that happens in "emergency mode" departments, agencies will generally also hold insurance specifically for that flow through that says "while the law says you can do anything, with due regard, in emergency mode, as long as you stay in our more restrictive SOPs, our insurer will also cover your personal liability".)