EMS always charges the person.
We often hear of people getting med-evac'd by helicopter from skiing accidents or getting lost on a hike. Those typically run about $30K or more, and often they aren't covered by insurance. A typical local ambulance run is about $500.
But those are private operations, it's different when the government is involved. I think most police/fire type operations have legal ability to seek some kind of payment but the grounds to so are really limited - not rescues based on accidents or ordinary negligence. That's just part of the taxpayer-funded service.
Of course, none of that is applicable here. But I don't think the rescuers are expecting any type of payment. Though if they were successful and did submit a request to recoup expenses, I suspect it would be happily paid. Not the governments (navies) though, I'm just talking about the few private orgs that are helping right now.