Yes they do. That seems to be a commonly misunderstood thing about the PJs. Note here I'm using "PJ" to refer to the whole operation, which includes the Pararescue guys, the Pave Hawks, C-130s, and all their support. In Alaska these are all components of the 176th Wing of the Air National Guard. The primary mission of the PJs is combat search and rescue, and in recent years at any given time a portion of the unit has usually been deployed "down range" in actual combat operations.

Like any combat unit, when not actually deployed in combat they are training for combat. Experience has shown that for their specialized mission of combat search and rescue, civilian SAR provides some of the best, most realistic training possible. Hence PJ units, both in the lower 48 and Alaska, intentionally allocate a portion of their training budget to civilian SAR missions. Thus civilian SAR is an intentional and integral part of their training plan.

Alaska is ideal in this respect. We have very difficult terrain, bad weather, and a very high frequency of missions. The air crew get to fly in very challenging conditions. The PJs get to keep their medical skills sharp by working on real hurt people, under true field conditions. About the only thing lacking from actual combat SAR is nobody is shooting at them. In my opinion it is a win/win for us taxpayers. The PJs get the most realistic training possible, and we civilians get our behinds saved!

Note that they don't necessarily accept any and all civilian SAR missions. For example for a purely medical mission, they won't take it if a private medevac company is available, capable, and willing to take the mission. Likewise for rescue missions, if the Alaska State Troopers can handle it using volunteer SAR teams the PJs will typically not get involved. And providing rescue support for other military training operations is always their priority over civilian SAR.