Since priests have advanced degrees in Theology as part of their formation they are uniquely qualified to teach Theology/Religion at the Catholic Schools and are often heads of the Department.
The Catholic Schools are generally run by the Diocese, unless they are a Private Catholic school run by a Religious Order.
It varies from diocese to diocese, and perhaps, from school to school. Keep in mind, though, that in addition to getting hired by a school, a priest would be required to get permission for such an assignment (a diocesan priest would need to receive permission from his bishop, and a religious order priest would need permission from his religious superior).
Can an priest from any of the Eastern Catholic Churches be able to teach at a Catholic school?
Yes, if he meets the particular requirements for teachers at that school.
Would they have to apply for bi-ritual faculties?
Bi-ritual faculties only apply to the celebration of the sacraments in the Latin and Eastern rites. If you’re asking purely about teaching at school, then no, bi-ritual faculties are not necessary to teach. (However, if a school is interested in hiring a priest, it’s not inconceivable that they’re interested, perhaps, in a priest who is also able to act in the role of chaplain, who would also be able to celebrate Mass and other sacraments for the students and staff of the school.)
We belonged to the NCEA. We never considered them unions. They were professional organizations. If I had a problem at the school, they didn’t necessarily help in any way. They were there mostly for accreditation. Not what is considered by laypeople as “unions” or mediators.
The State also accredits schools as does SACS.
But they do not have much jurisdiction over the Catholic schools. Catholic schools comply with their recommendations to say that they possess the accreditation because parents demand it. The Catholic schools feel like they have to compete with the Private secular schools and to justify the high tuition rates.
For example…our Bishop said that DRE’s do not have to have a teaching cert, but must have a Theology degree or similar. Having said that, the principals then decided that a teaching cert would look better on paper and got rid of many of the DRE’s who had been in place for years. The result was teachers who were well trained as teachers, but had little knowledge of their faith to pass on.
Generally a priest teaching at a school is assigned to that position by a superior. A parish pastor or curate in a parish with a school may take on some teaching duties there, like teaching catechism, for instance, but their duties remain primarily connected to the cure of souls.
For a religious priest, that’s a different story, however. Many religious orders are quite devoted to education and operate their own schools apart from the parochial school system. There, one might have a number of priests whose primary work would be teaching. But again, where and whether they would be teaching would be at the discretion of their superior.
As for Eastern Catholic priests, I don’t see any canonical reason why they could not teach, but the above restrictions would of course hold true as well. That said, though, I’m rather unfamiliar with the Eastern Catholic parochial school system or with EC teaching orders.