Paramedics would be with the fire dept., A public agency. Ambulance service is a private business. Some are under contract to the city or county, and will respond if requested.
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Is this common in every state?
NO! Here fire and ambulances are the same public dept. There are also private ambulance services, but they are mainly long distance transport that the public ambulances don't do unless emergency. Every paramedic here is also trained in fire. It is a requirement to become a paramedic. ENT's used in the public ambulance are generally trainees and after 2-4 years if they don't become paramedics, they will have to go work for a private company or find other employment. The first class paramedics here have at least a bachelors degree in emergency services, some a masters. Many are more knowledgeable of emergency medicine than the registered nurses in the emergency room.
Every response here whether ambulance or fire, requires at least one paramedic. The nature of the call dictates what trucks respond. If an accident or fire, the Chief or his Lieutenant are first with a basic tool truck. If a larger tool truck is needed, then he makes that judgement call. If the police are first to respond, then they make a judgement call, but always the Chief's rig is the lead and may call just a tool truck, the big fully equipped fire trucks, or maybe just the small fire truck. We also have a special Haz Mat truck that is a specialized version of the big tool truck.
It is interesting that all the different trucks have different siren sounds, with the firetrucks the only ones with airhorns. After 30 years living next to the station, I can tell who is responding without looking out. If I am outside, I always hold my right hand on my heart and my left in salute as they pass to show appreciation and say be safe. One of my neighbors, an old Japanese lady, holds her hands in prayer and bows slightly as they pass.
Fire and ambulance don't respond to just any accident like a minor fender bender. Who responds depends on the nature of the call. In domestic disputes, police respond and if mental illness or drugs are suspected, they call the ambulance that will have at least one paramedic, most times two and one maybe two ENT's. All paramedics here have emergency mental illness training.
If a smell that one reports as a possible dead body, then police are first with paramedics following.
The human dead body while having a distinct smell, does vary. One from gunshot or blade wounds will differ from one that OD'ed on drugs or alcohol. The alcohol OD has it's own distinct odor. The worse from my experience, is the smell of one burning alive. The last 58 years of my life have been affected by that smell. The only therapy that has helped me some, was hypnosis. All I can tell you is, it isn't a smell, a sight, nor a sound, you ever want to experience. Nothing is more horrible! I have PTSD from it and other things I cannot unsmell, unhear, or unsee.
Back to the OP, to sum it up, it all varies by state, county, and municipality. In Texas a game warden can stop and ticket you with speeding since they are state police. They may also be the first on the scene of a fire. The same here, but game wardens are so few and far between, that they will call in a speeder and continue on to where they were going. They will stop and help fight fire or save lives as these situations negate the urgency of their other calls.