Who discerns apparitions and private revelations?
From the 1978 document(unofficial english translation) from the Congregation for the Doctrine of the Faith on discernment of apparitions.
II. Intervention of the competent local Authority
1. As, at the time of a presumed supernatural fact, worship or an ordinary form of devotion is born in a quasi spontaneous way among the faithful, the competent ecclesiastical Authority has the serious obligation to inform itself without delay and to carry out a diligent investigation.
2. At the legitimate request of the faithful (when they are in communion with their pastors and are not driven by a sectarian spirit), the competent ecclesiastical Authority can intervene to authorize and promote various forms of worship and devotion if, assuming the criteria given above having been applied, nothing is opposed to it. But there must be vigilance nevertheless, to ensure that the faithful do not regard this way of acting as an approval by the Church of the supernatural character of the event in question (cf. above, Preliminary Note, c).
3. By virtue of his doctrinal and pastoral duty, the competent ecclesiastical Authority can intervene immediately of his own authority, and he must do so in serious circumstances, for example, when it is a question of correcting or of preventing abuses in the exercise of worship or devotion, to condemn erroneous doctrines, to avoid the dangers of a false mysticism etc.
4. In doubtful cases, which do not involve the welfare of the Church, the competent ecclesiastical Authority may refrain from any judgement and any direct action (more especially as it can happen that, at the end of a certain time, the supposedly supernatural event can lapse from memory); but he should not remain less vigilant about the event, in such a way as to be in a position to intervene with swiftness and prudence, if that is necessary.
III. Other Authorities entitled to intervene
1. The foremost authority to inquire and to intervene belongs to the local Ordinary.
2. But the regional or national episcopal Conference may intervene:
a) If the local Ordinary, after having fulfilled the obligations which fall to him, resorts to them for a study of the event in its entirety.
b) If the event assumes national or regional importance.
3. The Apostolic See can intervene, either at the request of Ordinary himself, or at the request of a qualified group of the faithful, or directly by virtue of the immediate right of universal jurisdiction of the Sovereign Pontiff (cf. above, IV).
Posted by Diane M. Korzeniewski, OCDS