Clearly, it is Jehovah Almighty who brings about healing (physical, emotional, and spiritual) on behalf of the sick. As David says in Psalm 103, "Bless the LORD, O my soul, and forget not all his benefits: Who forgiveth all thine iniquities; who healeth all thy diseases" (verses 2-3, KJV). So, God is the source of healing. But does that mean it is His responsibility to bring healing to the sick?
It seems to me that Jesus doesn't place the responsibility upon His Father to free the sick from their oppression (see Luke 4:18). Rather, Jesus set a precedent for His followers to go out and lay hands on the sick by doing so Himself (e.g. Matthew 19:15, Mark 6:5, Mark 8:22-25, Mark 10:16, Luke 4:40, Luke 13:13), by instructing His disciples to do so (cf. Matt. 10:8), and by prophesying that His followers would do so (Mark 16:18). Furthermore, we see that members of the early church laid hands on the sick (e.g. Acts 9:12, Acts 28:8), and Paul and James both instruct members of the church body to lay hands on the sick (1 Timothy 5:22, James 5:13-15), even calling it an "elementary teaching" (Hebrews 6:2).
So, whose responsibility is it to free the sick from their suffering? I'll let you decide that for yourselves.