It varies per county, but usually the state card is sufficient by itself. An example is Alameda County requires only the state card to work there, and if you wanted to work in Contra Costa County or Solano County, you could, however, if you wanted to work in Santa Clara County, they require ICS-100, ICS-200, NIMS-700, IS-3, AWR-160, IS-204, SEMS, county accreditation (wear a System ID Badge), and within three months of hire you need to attend a county orientation. Most companies require ICS-100, ICS-200, and NIMS-700 already, but in Santa Clara County, it's required by the county, not just the companies.
Your best bet is to look it up online, it's usually listed (e.g. in Santa Clara County, it's under Policy 214), ask the county EMS agency, or look up what companies in that county require you to have to work there. Sometimes stuff isn't required or can't be done until after hire (e.g. county orientation and System ID badge/accreditation couldn't be done until after I got hired at my current company).