As far as I'm aware there's no privates in house with LACoFD (except in La Habra, the Orange County city that contracts LA Co Fire...not OFCA for...reasons?....anyway my understanding is La Habra itself contracted Care to staff their 2 ambulances and they're in house with LACoFD and dispatched on their dispatch but only for calls in La Habra)
But besides those two units, I'm not aware of any official in house (beyond "go post at this fire station and the crews will let you hang out inside") but yeah "fire shift" meaning running with fire (who is in charge of all 911s in LA and OC so fire shift is synonymous with 911 shift).
At least with McCormick every shift is a "fire/911/IFT" shift as every unit is up for 911 calls, and even the odd IFT thrown your way though I've gotten maybe a half dozen BLS IFTs while on a 23hr shift in the year or so I was there? The 12hr day car shifts that are considered auto first up for all calls they're in range of prob get a few more...but I can't comment for sure on them (depending on the ALS contracts at the time the medic units will either be running those all day or be sitting in the 911 rotation like a 'normal BLS unit)