one is supposed to get permission from one's bishop to change rites, though I doubt that any laymen bother with that.
Although shawn's question dealt with bi-ritual clergy, Rr's opening remark in response speaks of lay persons, so let's deal briefly with that first.
Lay persons are not, cannot be bi-ritual (a term one occasionally, and erroneously, hears applied to them). A lay person belongs to/is ascribed to a single Church [sui iuris], regardless the Church in which he chooses to worship.
A lay person does not 'change rites' (that terminology is antiquated and has not been proper for several decades now - not since the Eastern and Oriental Churches were formally recognized as distinct Churches sui iuris
(Churches of their own law).
A lay person transfers from one Church sui iuris
to another (e.g., the Latin Church to the Byzantine Ruthenian Church or, more rarely, vice-versa). The act of doing so is termed a Transfer of Canonical Enrollment and is accomplished by petitioning one's current bishop and the bishop of the Church to which one seeks to transfer.
It is not necessary to formally seek such a transfer merely to worship in and receive certain Mysteries within a different Church sui iuris
than the one to which you belong - because a Catholic may worship in and receive certain of the Mysteries in any Catholic Church. However, if one wants to have their children receive the Mysteries of Initiation (Baptism, Chrismation, and the Eucharist) in a Church other than their own, or if one intends to pursue the Mystery of Holy Orders in a Church other than their own, a formal transfer should be (in the former instance) or is (in the latter instance) required. Additionally, many persons seek to formally transfer because they believe that they can best live their spiritual life in a different Church than the one to which they are ascribed.
There are also specific provisions related to entering into Marriage with a Catholic of another Church and with respect to the children of one who formally transfers. I'm neither going to address those here nor delve into any more detail on the subject of Transfers of Canonical Enrollment for laypersons here, as the topic is extensively dealt with and described in multiple threads in this forum. (If anyone needs information on those, any of several of our members will be pleased to provide links to relevant threads.)
However, I want to dispel any notion that laypersons 'do not bother' with the process; many have and do - including a not insignificant number of our forum members.
Probably these priests convinced their bishops to permit it, or perhaps since the general relaxation of discipline after Vatican II a priest can just do it.
No and no. A priest does not convince his bishop to permit him to obtain bi-ritual faculties (the proper terminology for what the priest is granted). Nor can a priest 'just do it'!
But then again, all priests, to be licit, must be attached to a diocese or an eparchy. They probably have to get permission from both the RC bishop and the Bishop or Metropolitan of the relevant Eastern rite, and they probably have to live in a part of the world where both East and West share territory.
Yes. For a priest to be granted bi-ritual faculties there must be agreement on the part of his bishop or his religious superior (in the case of priests who are of religious orders, rather than diocesan/secular priests) and of the bishop of the canonical jurisdiction for which they seek faculties. (Rr's comment presupposes that all such grants of faculty are made to priests of the Latin Church to serve bi-ritually in an Eastern Church. Such is not the case, there are priests of the Eastern Churches who are granted faculties to serve in the Latin Church or to serve in a different Eastern Church.)
The request to grant bi-ritual faculties to a priest (in either direction) is made by one bishop to another.
A Latin bishop whose canonical jurisdiction includes some meaningful number of Eastern Church faithful is instructed by Canon Law to provide for their pastoral care. Accordingly, he may reach out to his brother bishop of that particular Church and ask that a mission be opened within the geographic area of his jurisdiction - and that a priest be assigned to it.
If the Eastern bishop agrees but does not have a presbyter to staff it, the Latin bishop might seek out a priest of his diocese with an interest in receiving bi-ritual faculties and offer his services. Subject to the Eastern bishop considering the priest acceptable, he will arrange for the Latin priest's instruction and grant him faculties.
In the case of an Eastern priest who is serving a small mission or parish in an area where there is a priest shortage, a Latin bishop might reach out to that Eastern priest's bishop and ask if he would consider allowing him to grant the priest bi-ritual faculties in the Latin Church, so that he might assist at a nearby Latin parish or in some chaplaincy role. In most such cases, the Eastern bishop will agree to this. The added income in such instances will often help to offset the salary and other financial needs of the Eastern priest and his mission or parish.
In such instances, the bi-ritual priest remains incardinated to the diocese or eparchy of his origin or to his religious order and - particularly in the case of diocesan or religious order Latin priests with bi-ritual faculties - the time may come, often does, when the priest is needed elsewhere and may have to be pulled back by his bishop, leaving a gap to be filled or a mission to be closed/parish to be suppressed.
Deacon Richard, in his post above, pointed out that Eastern Catholic clergy serving as armed forces chaplains are required to obtain bi-ritual faculties in the Latin Church. The same is true of those serving as chaplains to Veterans Administration facilities.
As regards religious orders, there are a number of Latin religious orders which have a history of involvement with the Eastern Churches, including some which have specific provinces or other organizational entities within their structure which are solely dedicated to to the Eastern Churches.
Priests of the Society of Jesus who are specifically ordained for the service of an Eastern Catholic Church [vs Jesuits who were initially ordained as Latins and subsequently accepted to the Society's ' Eastern Province'] are ordained in the Rite of the particular Church which they are to serve, but are required to subsequently be accorded bi-ritual faculties in the Latin Church, although they may never serve it.
The same may also be true in the case of at least some other principally Western religious orders that have defined provinces or sections that serve in Eastern Churches (Benedictines, Franciscans, Marianists, Assumptionists, Salesians, Capuchins, and Redemptorists come to mind).
As regards the non-Byzantine Churches:
The vast majority, if not all, priests of the Ethiopian Catholic Church are accorded bi-ritual faculties in the Latin Church (and the obverse is true of Latin Ethiopian priests). It goes hand-in-hand with the unusual set-up of canonical jurisdictions in Ethiopia - where those of the Ethiopian and Latin Churches are exclusive, rather than overlapping and each must, of necessity, afford pastoral care to the faithful of the other Church who reside outside the geographic bounds of their own jurisdictions.
As regards priests of the Syro-Malabar Church, the majority also hold bi-ritual faculties in the Latin Church. The very significant number of priestly vocations in that Church have made it a ready source for priests to be 'loaned' to Latin jurisdictions, not only in India, but elsewhere.
I don't know that the numbers are substantial, but I have known several Maronite priests with bi-ritual Latin faculties.
As far as I know, it's not particularly common among the Coptic, Syriac, Armenian, Chaldean, or Syro-Malankara presbyterates to have Latin faculties.
However, there are other combinations to be found. One can find priests of the three Antiochian Patriarchates (Maronite, Melkite, and Syriac) who have faculties in one or both of their Sister Churches.
A few years ago, Deacon Randy mentioned a priest of his acquaintance with faculties in 4 Churches besides the Byzantine Church to which he was ordained.
Father Archimandrite James King, of blessed memory, a Melkite priest, had both Armenian and Latin faculties and has also, at various times, assisted in/served parishes of three other Byzantine Rite Churches - Ukrainian, Russian, and Ruthenian - besides his own.
There are some Malankara and Malabarese priests with faculties in each other's Churches. In particular, the Syro-Malabarese maintain a vicariate comprising 15 parishes in one of their eparchies for Syro-Malankara Knanaites, in which the liturgical forms are those of the Malankarese's ancestral (Antiochene) Rite (despite that the Malabarese Church is of the Chaldean Rite) employing a Knanaite Usage (most Knanaites are Malabarese).
For a priest of any Church to effect a Transfer of Canonical Enrollment requires the permission of the Oriental Congregation - which will typically be granted if there is no objection on the part of the losing hierarch (or religious superior if it's a religious order priest) and the gaining hierarch. Most commonly, this occurs with a priest who has been serving for a protracted period in the Church for which he was accorded bi-ritual faculties
Bi-ritual deacons seem theoretically possible, but the office of Deacon is not the same in East and West. An Eastern Deacon has a larger role in the Liturgy--I don't think that an Eastern Deacon would want to be bi-ritual, and a Western Deacon would not be qualified.
Again, as I have indicated before, disparagement of the Latin Church or its Liturgy or clergy, by insinuation or otherwise - such as dismissive commentary regarding the Latin diaconate - is unacceptable in this forum!
Deacons do not technically receive bi-ritual faculties. With the permission of his own bishop, a deacon may be given a blessing by a bishop of another Church to serve in the latter's jurisdiction.
Examples of several sorts come immediately to mind: our own Deacon Ed Faulk, was ordained a Latin deacon, but serves a Melkite parish; Deacon Stan, of blessed memory, (husband of our own Pani Rose), was ordained to the Byzantine Ruthenian Church, but served a Melkite parish throughout his diaconal career; Deacon Lawrence Cross of Australia was ordained a Melkite deacon, but served a Russian parish prior to his presbyteral ordination (and now does so as a priest); Deacon Gregory Haddad, a Melkite, serves the Ruthenian mission in New Orleans; and, at least one Melkite deacon whose name escapes me presently serves a Latin parish, due to there being no Byzantine parishes in the area to which he had to relocate for employment reasons.
These are just guesses, I am no expert.
Guessing is not the ideal - virtually everything posted above can be found by judicious use of the search function.