Any priest with faculties can confirm any person privately or receive a person into the faith, giving the sacraments of baptism, confirmation, penance and Holy Eucharist, any time he sees fit. . . .I have been to a wedding where the priest confirmed the groom prior to vows… . …
As a general statement, the first statement is not correct, and one has to be careful because the validity of confirmation is at stake. In the Latin Church, the valid confirmation of a person by a priest requires the existence of a faculty according to canon 882: “The ordinary minister of confirmation is the bishop; a presbyter who has this faculty by virtue of either the universal law or a special concession of competent authority also confers this sacrament validly.”
Provision is then made in canon 883:“The following have the faculty of administering confirmation by the law itself: 1º within the limits of their territory, those who are equivalent in law to the diocesan bishop; 2º with regard to the person in question, the presbyter who by reason of office or mandate of the diocesan bishop baptizes one who is no longer an infant or one already baptized whom he admits into the full communion of the Catholic Church; 3º with regard to those in danger of death, the pastor or indeed any presbyter.”
Thus the situations that Grandfather relates strike me as most understandable on the basis of this provision rather than on the notion that a priest could do these things as he sees fit.
The mention of the case of Easterns though would involve additional consideration.
Canon 696 of the code of canons of the Eastern Churches provides "§1. All presbyters of the Eastern Churches can validly administer this sacrament either along with baptism or separately to all the Christian faithful of any Church sui iuris including the Latin Church. §2. The Christian faithful of Eastern Churches validly receive this sacrament also from presbyters of the Latin Church, according to the faculties with which these are endowed. §3. Any presbyter licitly administers this sacrament only to the Christian faithful of his own Church sui iuris; when it is a case of Christian faithful of other Churches sui iuris, he lawfully acts if they are his subjects, or those whom he lawfully baptizes in virtue of another title, or those who are in danger of death, and always with due regard for the agreements entered between the Churches sui iuris in this matter.
However, when a Latin Rite priest baptizes the child of Eastern Catholics, the child becomes a member of the Eastern Church sui iuris nevertheless. Since the law of the Latin Church did not provide this priest with the faculty to also confirm the child, Chrismation with Holy Myron would have to be provided later.
(In passing, keep in mind that when a Latin Rite priest receives a member of an Orthodox Church into full communion, that person becomes a member of the corresponding Eastern Church sui iuris and not the Latin Church. They would have been, of course, Chrismated and Eucharistized already. Now as canon 35 of the Eastern Code says: “Baptized non-Catholics coming into full communion with the Catholic Church should retain and practice their own rite everywhere in the world and should observe it as much as humanly possible. Thus, they are to be enrolled in the Church sui iuris of the same rite with due regard for the right of approaching the Apostolic See in special cases of persons, communities or regions.”)
While the conferral of any sacrament has a necessary public character, it is possible that only a small group gathers at the event and that it is not publicized to broader community of the Christian faithful. In that sense we may speak of “private” Baptism or Confirmation or reception into full communion.