Healing… before the Cross/Resurrection:
In the Gospels, before the Cross, healing was apparently used mainly as a tool of evangelism, to draw the people and convince them of the power in Christ, His divinity. Christ, Himself, healed ALL who came to Him. In addition, He gave each of the twelve disciples the gift of healing as part of their evangelical mission.
Christ healed everyone (Matt 4):
And Jesus was going about all Galilee teaching in their synagogues, and proclaiming the good news of the reign, and healing every disease, and every malady among the people,
and his fame went forth to all Syria, and they brought to him all having ailments, pressed with manifold sicknesses and pains, and demoniacs, and lunatics, and paralytics, and he healed them. Matthew 4:23-24 Young’s Literal Translation (YLT)
Disciples were all given healing gift (Matt 10):
And having called to him his twelve disciples, he gave to them power over unclean spirits, so as to be casting them out, and to be healing every sickness, and every malady.
`And, going on, proclaim saying that, the reign of the heavens hath come nigh;
infirm ones be healing, lepers be cleansing, dead be raising, demons be casting out — freely ye did receive, freely give. Matthew 10:1,8 (YLT)
Healing… after the Cross/Resurrection:
After the Resurrection, in the new, growing church, it is apparent the gift of healing was given to select, but not all believers, as demonstrated below in 1 Corinthians 12.
for to one through the Spirit hath been given a word of wisdom, and to another a word of knowledge, according to the same Spirit;
and to another faith in the same Spirit, and to another gifts of healings in the same Spirit;
and to another in-workings of mighty deeds; and to another prophecy; and to another discernings of spirits; and to another [divers] kinds of tongues; and to another interpretation of tongues:
[are] all apostles? [are] all prophets? [are] all teachers? [are] all powers?
have all gifts of healings? do all speak with tongues? do all interpret?
and desire earnestly the better gifts; 1 Corinthians 12:8-10; 29-31 (YLT)
In modern day, whether in person or through the testimony of documentaries such as those made by filmmaker, Darren Wilson [Finger of God, Furious Love, Father of Lights, etc], we can see evangelical teams witnessing on the streets around the world and healing the sick and lame. There are times not everyone is healed, however. It seems to be determined by the will of the Spirit for His purposes.
Yet, it also seems there are some unique individuals who are able to heal every person they lay hands on. This would indicate they have the gift of healing, specifically.
Perfectly in line with Scripture, we might conclude God can use anyone to heal when He wishes to reach the lost. However, some in the body of Christ possess the gift of healing to benefit both the body of Christ and non-believers.