I do BLS IFT. Depending on who gets there first, the first person to get there usually does a rig check: make sure everything is in the rig, write down notes that we are required to report like milage, and clean the inside and outside of the rig. Usually while they are in the middle of cleaning the rig, the other partner gets there and starts helping out with cleaning. If the other person gets there around the same time, they'll divide up the cabinets and verbalize the info that needs to be recorded e.g. person in front will say the milage and person in back will say the gurney, main O2 tank psi, and the medical air psi. Whoever has the clip board will write down what was verbalize.
We'll ask "Do you want to tech or drive the first call?" Every call after that, we'll trade off UNLESS it's beneficial for the person to the tech/drive the call instead e.g. if the patient is a Spanish-only speaker and the person is suppose to drive speaks Spanish/person that is teching can only speak English. We'll trade off then since it's better for the patient.
When we go into the hospital, clinic, or home to transport the patient, the person that techs the call grabs the paperwork, reviews the chart, and gets a report from the nurse. The driver is gonna take the gurney to the patient room, introduce himself and the partner/person that's gonna tech "Hey, my name is Andrew. I'm the driver. My partner name is Dahif, he's gonna be in the back with you today." He'll assess how oriented the patient is (ask AO questions), get baseline set of VS, and if possible, do an assessment of the patient (ask if there are any complaints, S/S, ask about when was the last time they ate, if they have any belongings, etc). If the partner isn't done with the paperwork, the driver will return to the tech, and report all of his findings. If the patient is able to ambulance on his own or with the assistant of one person, he'll get the person onto the gurney, strapped, and ready to go (5150 calls are usually like this).
After the patient is tranfered to the receiving facility, the tech is gonna get signatures/report to a nurse, and the driver is gonna take the gurney back to the rig, clean the gurney, rig, make sure everything is ready for the next call. If the tech returns to the rig after finishing the report, he will usually help the driver prep the rig for the next call, and either finish his PCR during down time after the call, or if there is another call/it's busy, finish his PCR on the way to the next call. Sometimes the driver will relieve the tech of helping with cleaning and ask the tech to finish his paperwork while he cleans.
At the end of the day, both people will make sure that the rig is full of gas, stocked, clean, and ready for the next crew. Sometimes we'll prefill the paperwork for the next crew regarding milage, O2 tank, etc. The next crew is still responsible for all of this though/it would be risky if they didn't double check if something wasn't missing or a number was off. Some of it can be handy like we have to write the VIN, license plate, etc, so that can definitely make things easier for them.
Unfortunately it doesn't work like that with all crews, and some of my partners haven't done it this way, but I try to go by the above format usually. Sometimes I'll get an ambulance that's totally trashed, gas isn't full, isn't stocked, work with partners who doesn't want to hear the driver assessment, doesn't help clean the rig after a call, etc. This is just the format that I've mostly gone by, and try to go by.