I believe that healings are possible. A friend of mine had one: He had a wart on his finger and someone prayed over it and it fell off. He was quite shocked, because he's Japanese and not Christian. (I wasn't there, but he told me and he's a banker, not a person who's into airy-fairy stuff.)
However, I don't think it's, like, one person has the power and zaps another, or that God says, "Bingo!" for one person and not another. I think a person CAN be a conduit for God's healing power and another can be receptive to it. (Remember that passage where Jesus said he'd felt power go out of him, when the woman with a flow of blood touched him?)
As to how it operates, I think we need to form a less ham-handed understanding of the relationship between matter and spirit. Dutch says, "Process thought sees that all things, from atoms and cells to humans, are related externally and internally. " I think you have to go to that level to make sense of it and we don't really understand enough about it right now.
I do remember going to healing services regularly at one time in my life (for emotional rather than physical difficulties). When they would lay hands on my head, I usually felt waves of heat go down through my body. You can make up all kinds of reasons about why you think that happened, but for me, it simply happened. For that matter, I've had physical problems cured by acupuncture, and we don't know how that works, either.
So yes, I believe there's such a thing as healing (whether "faith healing" or "energy healing" or anything else), but I don't believe it's either arbitrary magic zapping OR just a placebo effect. Science is great, but it still can't properly account for near-death experiences and it's helpless with the non-material.