I believe that pastoral care is the responsibility of the whole church community, but the minister has prime responsibility for this. We are in community together, and as a community, we must care for one another. That said, there are times when the minister's special training and experience is called for.
I received my most intense pastoral care training during my hospital chaplaincy internship, where I was responsible for the emergency department, intensive care unit, and maternal and child health, in addition to covering the entire hospital when I was the on-call chaplain. I continued to work as a per diem on-call chaplain following my internship.
I was among the Protest Chaplains during the Occupy movement, and I set up and administered the Facebook page for the Protest Chaplains DC. During Occupy DC, I pastored to protestors and police alike, and held impromptu worship services outdoors.
As a mission chaplain in the Civil Air Patrol, I am, from time to time, called upon to provide emergency pastoral care services as needed, in addition to the services that I provide to the squadron that I serve on a regular basis.
Whether in the church, in the community, in homes, or in hospitals, it is possible to minster to others. Ministry can often happen when we least expect it. This is why I will sometimes have community office hours in local coffee shops - to allow people to meet with me in a more informal setting. Caring for an entire church community still happens most often one-on-one.