The sacrament of anointing, because of its association with the forgiveness of sins as well as medieval/Tridentine tradition, may be conferred only by a presbyter or bishop. In the immediate post-conciliar years, there were rumblings about deacons and lay people serving as extraordinary ministers. There is still a need in some hospital settings and elder care facilities, especially in rural areas or mission lands for a more prompt celebration of anointing.
The drawback to the current practice is that anointing becomes like confirmation in practice: a sacrament accommodated to the schedule of the clergy rather than the need of the people.
That said, the authoritative confirmation of priest-only anointing came at Trent. In the first millennium, though, it is well-documented that lay people anointed the sick:
MINISTER OF THE ANOINTING OF THE SICK
16 The priest is the only proper minister of the anointing of the sick. (See Council of Trent, sess. 14, Dc Extrema Unctione, cap. 3 and can. 4: Denz.- Schön. 1697 and 1719; see also CIC, can. 1003)
This office is ordinarily exercised by bishops, pastors and their assistants, priests who are responsible for the sick or aged in hospitals, and superiors of clerical religious institutes. (See CIC, can. 1003)
Deacons and lay people may assist in the preparation of the sick. Note also the rite recognizes the need for preparing the loved ones of the sick person, as well as medical personnel:
17 These ministers have the pastoral responsibility both of preparing and helping the sick and others who are present, with the assistance of religious and laity, and of celebrating the sacrament.
The bishop has a responsibility for communal celebrations across parish boundaries or at care facilities. Some parishes take pastoral responsibility for hospitals and elder care facilities within their boundaries. There’s always a discussion about non-parishioners who maintain membership in their home parish. If there are numbers of such people, who’s responsible: the bishop or the pastor? Most bishops delegate by default, if they consider the situation at all.
The local Ordinary has the responsibility of supervising celebrations at which sick persons from various parishes or hospitals may come together to receive the sacrament.
Clergy keep lines of communication open and work together:
18 Other priests also confer the sacrament of anointing with the consent of the ministers mentioned in no. 16. Presuming such consent in case of necessity, a priest need only inform the pastor or hospital chaplain later.
19 When two or more priests are present for the anointing of a sick person, one of them may say the prayers and carry out the anointings, saying the sacramental form. The others may take the remaining parts, such as the introductory rites, readings, invocations, or instructions. Each priest may lay hands on the sick person.
Any thoughts, especially about the need for extraordinary ministers of anointing?