What happens when someone lawfully in the USA triggers a border rescue beacon?

I'm going to take this form the point of view of a generalized Search and Rescue as I feel it is the equivalent of someone activating a personal Locator Beacon in order to call someone to come and get them.

From some quick googling, who foots the bill for SAR differs depending on what state you are in. In a lot of states it is free. But in other states you can be charged - because the state wants to reduce the number of people taking a risk and reduce the cost of SAR operations (One example I've seen is skiers deliberately skiing out of bounds being charged)

I found this blog from 2018 that is Arizona specific. It says in part:

In Arizona, a state fund covers 50% of the first $1000 cost of a rescue, 75% of the next $20,000, and 100% of the rest.

So it looks like you could be up for a $5,000 bill in AZ.

Also see this Great Outdoors question Who gets the bill if you activate a PLB to help someone else?

Here is another perspective on costs in general

In the U.S., whether you have to pay depends on where exactly you are when you get into trouble. In any of the national parks, the government picks up the tab for your rescue. The National Park Service spends nearly $5 million annually on search and rescue (SAR) missions and that doesn't include the cost of hundreds of thousands of man hours that go into these searches. Yet unless rescuees violated a park rule — like trespassing into a protected archeological site, for example — they aren't responsible for the cost.

...

In Wyoming’s Teton County, home to Jackson Hole resort, the search and rescue crew works in conjunction with the county sheriff. Each year, they conduct an average of 70 rescues, according to Doug Meyer, the area's SAR coordinator. And even though most of the rescuers are volunteers, costs can still add up for equipment and resources — such as leasing a helicopter, and maintaining ropes and radios. "We only get back about 20% of that cost," Meyer estimates. They do charge for helicopter flight time, which runs at about $1,600 per hour, but there is no strict enforcement of payment

...

In New Hampshire, officials are already doing just that. A decade-old law charging people for the costs of their rescue if the behavior that got them into the mess was deemed "reckless" was rewritten this past summer, lowering the bar so that merely "negligent" behavior could saddle you with a bill.

FYI I carry a PLB when I'm the wilderness, and if I need to use it, I'm activating it and not even thinking of the cost.