We do have an independent Mountain Rescue Association here that can certify teams for specialized technical rescues, and those teams are generally among the best in the country. Maybe not skilled to alpine mountaineering standards, but they can search for and retrieve people from extremely challenging terrain at 14,000 ft.
There is also a national search and rescue organization (NASAR), but they are a joke, speaking candidly. They have almost no presence in the west and we can't figure out what they're actually doing with the membership dues from the few suckers that decide to join in exchange for a sticker. They offer some certifications in some regions, but they're optional and many teams just develop their own programs.
But, nobody gets to deploy until a sheriff says-so, teams usually need a sworn (law enforcement) member escort on any operation, they are funded by their local sheriff's office (and whatever grants or community donations they can get), and the sheriff can decide when it's time to go home. That makes SAR across the US a patchwork quilt of solid teams and ragtag posses, with a broad variety of attitudes and standards.
There is a mutual aid system that allows counties to develop agreements (MOA/MOU) to render aid if requested, and is often used to dispatch higher-qualified resources to difficult operations in poorer counties.
But it's still, always, a law enforcement operation.
If they need a specific tool or capability only the police or the firefighters have, they ask them for help. That being said those alpine units are closer to an ambulance than to law enforcement, because that makes more sense