That's is a great question. Actually we have a reference in the New Testament about it. The apostles came to Jesus saying how other men were 'healing' and kick out demons from other people, the apostles were complaining because these guys didn't have the 'authority' to do so. I don't remember the exactly words of JesusChrist but basically he told them that whoever do those kind of things in his name, it's good.
I do believe personally that there is people out there who don't have the authority to heal but they do it through the FAITH of the person who is sick. Faith cam moves mountains and if the intention it's good then it comes from God. Let's clarify that a person doesn't have the power to heal but it's the faith of the receiver that the miracle is performed.
Joseph Smith taught in the Teachings of the Prophet Joseph Smith that women could also lay their hands on the heads of the sick and heal them. In the early days of the church certain women were given the power to heal from priesthood leaders so that they could anoint the sick and pronounce a blessing on the sick.
In order to be more in tune with the revelations though, I think it is safe to point out that God has said that when there is a sick person we should call the elders and for that reason the church has gone away from the practice of women anointing the sick. I have heard of it being done even today but nothing official.
One interesting thing to note is... when an Elder gives a blessing the words are... "WE bless you" or "I bless you" whereas in a prayer we address Heavenly Father. In a blessing we address the person and proclaim what will be done through the power of the Priesthood and Jesus Christ, in simple prayer we plead with Heavenly Father for what we would like to have happen.
I remember that Hyrum Smith's widow blessed her ox that it would recover and pull their wagon to their destination. There was a time while Brigham Young was president that he spoke in a Conference about the sisters who were out giving blessings in the valley. He told them that they needed to state their priesthood authority as "the priesthood given to them in the temple endowment" then they could go ahead and give a blessing. Today that type of action is highly discouraged and not talked about as though it had never happened.
I was hoping to keep this Topic about those who are not Members or Priesthood Holders who claim the power to heal as in healing hands, etc. The Sisters exercising Priesthood is something we share in the Mature LDS Board only since going into the Temple is not for Public Discussion. More on that can be found here: Source 2 in four pages of Discussion..