This is true - they were always on duty
I realized this watching s6, ep 24 last night, the scene switched from Johnny, who was up late writing his game show script, to Rampart Hospital where it's 2 a.m. and Dixie and Dr. Early are still working.
The engine and squad get called out separately quite often, and the paramedics usually stop active involvement in cases at the hospital doors. In a few instances, Roy and Johnny start their shift and the squad is just coming back from another call with another pair of paramedics just ending their shift, or they end their shift and hand over to another pair to take over the Squad, and the Station 51 crew is often shown coming on or going off duty as part of the firehouse plot. However, it's played straight with the hospital; it seems that no matter what time of day or night it is, Brackett, Dixie, Early and Morton will always be there. - TV Tropes
