The answer to your question comes down to two things - veracity and trust. I have talked to enough people who have seen or experienced miraculous healing that it would be unreasonable for me to start with the assumption that miracles don't happen. Some people might ask for proof. The problem with this position is that unless you are an eyewitness to the miracle, you really have no way to verify the veracity of the event. An eyewitness can say, "I saw it happen!" If you choose to not believe their testimony, then the issue is no longer about whether the miracle happened, it now comes down to trust.
If you start from the position that miracles don't happen, you won't trust any eyewitness to a miracle by definition. The end of that dialog can only go one way: "Hey!! I saw a miracle." The skeptic answers, "No you didn't, you liar." And that is the end of dialog. As soon as you call someone a liar, there is nothing else to say. Of course, some people DO lie, but even then, you still have the question of how to verify.
You can answer this question more broadly. What happens when a person "gets religion," or "gets saved," or "finds the Truth." A person who doesn't believe in that religion will respond, "Oh no you didn't." The problem with skeptics who reject everyone's religious experience is that they don't understand that denying a person's experience is not proof that it isn't true. A skeptic's lack of faith or denial of faith is a personal position that is not an argument for or against anything.
Also, if someone came up to me and told me they just had a profound and deep experience of the miraculous, I would believe them because it would be nasty to throw cold water on their experience. If their experience was not true, that will show itself in time. In the meantime, why would I deny someone their deep religious experience.