How do believers in present-day miraculous healings respond to skeptics requesting thorough scientific medical documentation accessible worldwide?

God's purpose in providing miracle

As one of those "believers in concrete present-day cases of miraculous healings" I would respond to the skeptics that the purpose of miracles for today is the same as the purpose of miracles that God caused through the apostles & first generation believers (including those NOT recorded in the Bible): to help people trust Christ and become Christians. For those who are already Christians, the purpose is for them to stay Christians (let's say they are in a crisis of faith needing a sign that God hears their cries) or to grow in their trust in the Lord so they have more confidence to risk their lives more for missions that the Holy Spirit sends them to (such as being a missionary in a foreign land or to confront ungodly powers like Putin).

In the Bible, miracles always carry a purpose. Many in 1 Kings is to authenticate a prophetic message to kings. Many in the Gospels is to authenticate Jesus as true prophet as well as evidence to authenticate the arrival of Jesus's kingdom breaking into the world of which we can be a part of, by being a member of Christ's body. So the level of evidence that God providentially provides can arguably varies depending on what the recipient needs to realize the Holy Spirit's purpose. Some people need a certain kind of evidence; others need another kind.

God's prerogative in setting the type/level of epistemological justification

The primary beneficiaries are NOT third parties like us, researchers, or even well meaning theologians like Craig Keener. So we cannot expect a uniform level of "epistemologically justified" evidence as though God is making himself available to be measured scientifically! When I read 1 Kings today it struck me how the narrator casually show how the evidence level varies according to the person the Lord wanted to show Himself to. In call cases, it is a personal message to the primary witness albeit through the language of power, from a very public humiliation of the prophets of Baal (definition #2 first degree miracle of fire licking water in 1 Kings 18:38) to a very private message of judgment (something that can easily be mistaken as a coincidence when the sick child dies as soon as Jeroboam's wife walks through the door in 1 Kings 14:17). It makes sense that God used a degree 1 miracle for the public one, but not even something that could register as a miracle (of definition #2) for the private one.

Then there's the hardened heart type of recipient I refer to in my other answer to whom God probably wouldn't even bother to provide a miracle. Maybe that's why to the Pharisees who demanded a miracle to be performed RIGHT THEN in their presence (presumably when they have all their "measurement equipment" ready) Jesus refused the request by saying that it was sufficient that the sign for them would simply be the sign of Jonah (Matt 12:38-41).

In other words, I will say to those skeptics that "the assumption that God will always ensure providentially that such level of evidence will be available to validate each and every one of the miracles He ordains, so that belief in such reports will be epistemologically justified" is a false assumption, or at best the epistemological justification type/level depends on what the targeted beneficiary needs! To confront hardened hearts of King Ahab, Queen Jezebel (who sponsored those Baal prophets), yes, a publicly visible and EMPIRICALLY VERIFIABLE miracle of definition 2 degree 1 was called for. Even so, Queen Jezebel didn't repent, while King Ahab is portrayed in 1 Kings as an inconsistent guy who is too much controlled by his wife but still has a little respect for the God of Israel. But for Jeroboam's wife who was delivered a prophetic message earlier in the day, she wouldn't need THAT kind of epistemological justification to be convinced that the God of Israel was truly who was behind the death of her sick child. We are not even told whether Jeroboam's wife repented or whether she saw it as more than a coincidence, but at least for the narrator of 1 Kings, the event was epistemologically justifiable enough even though it's not a definition #2 miracle. I know this question is about healing while I cited a death "miracle" but the point is to show that God is the master of life and death and can heal/kill at will and at the time of his choosing.

Conclusion

Therefore, I would say to those skeptics that the God behind healing miracles today provides a variable type/level of epistemological justification to the primary beneficiary. I know a friend whose mom became Christian (this is in a non-Western country in the 1970s) because after a Christian prayed for her, she was cured from some debilitating internal disease that the doctors gave up, that they thought would soon killed the mom. I don't have the details, so we don't know whether this is psychosomatic, wrong prognosis (she wasn't really about to die), unavailability of Mayo clinic level of treatment in 2022, degree 3 or degree 2 miracle, etc. BUT the epistemological justification type/level that God tailored specifically to my friend's mother (and to my friend also, who was in her elementary school years), which (to my knowledge) didn't include medical test results, was sufficient to them. The bottom line for them was: the mom was almost dead, a Christian prayed over her, and soon after she could live (for about 40 more years). That event (yes, they called it a "miracle" although we may not agree with them) was sufficient to move their hearts to say that the "miracle" came from God who loved them and who wanted them to be Christians for their own good. In my view, the Holy Spirit successfully realized His purpose in using a "miracle" to bring another lost sheep to the fold.

Thus I think it is reasonable to theologize (for I dare not to be more certain than mere theorizing) that in determining the level and type of epistemological justification what matters to God is the skeptics's own openness to Him and their motivation for wanting miracles as a sign. God knows their hearts and knows whether they need one, and if so, the timing. God provided one to C.S. Lewis through a very personal circumstances when C.S. Lewis out of love to Joy his beloved offered himself as substitutionary sacrifice when he cried out to the Lord, and was granted a temporary remission of bone cancer in April 1957 (with possibly X-ray evidence) but which 2.5 years later came back and ultimately killed her. How C.S. Lewis processed this event was described in page 77 of a Mythlore 35.1 (Fall/Winter 2016) journal article "It Was Allowed to One": C.S. Lewis on the Practice of Substitution citing a November 1957 letter he wrote a few months after the remission to Sheldon Vanauken (whose wife died only in her 40 after a strange illness in January 1955), probably in C.S. Lewis effort to help Vanauken with his deep grief, which he later wrote in his 1977 book A Severe Mercy.