For property in unincorporated LA County, LA County Public Works issues building permits, and its list names new building, alterations, repairs and grading, not concrete cutting. That doesn't settle whether your job needs one, and property outside the county's area may go through another office. We check with the right office for your address before we set a date, and when a permit's needed we take care of it and the inspections.
A cut that reaches into the street or a public sidewalk is a separate matter, because the county's Public Infrastructure team handles permits for road work, encroachment and digging in the public right-of-way, meaning the public streets and land. Around here a lot of what we cut is older driveway and patio slab on tight lots, where the line runs close to a house wall or a fence.
A California safety rule requires a dust reduction system any time powered tools cut, grind, core or drill concrete. The reason is the fine silica dust a saw throws off, which you can breathe in and which can cause silicosis, an incurable lung disease, and lung cancer. We do the same work in Montebello, Monterey Park and plenty of other cities nearby, where most cities run their own permit counters.