If you read the Pillars of the Earth, (Ken Follett), you would certainly have been left with the impression that the "Mother Church" footed the bill for the local congregations.
In actual fact, Cathedrals, which often are the seats of Bishopdoms, are paid for with pooled funds, but that is rarely the case for local facilities. There are some mission churches around.
'Pillars of the Earth' by Ken Follett is a masterpiece! It is a long book that takes little time to read it, if you do as I did - read it in long, continuously intermittent sessions - it is hard to put down once one starts reading.
As for ownership, Cathedrals in Ireland changed hands regularly for a while, when Ireland was all a part of Great Britain and when the church switched back and forth between Roman Catholic and Church of England (or Ireland). The ancient Cathedral of St Patrick (Seat of the Church of Ireland Archbishop of Armagh, Primate of all Ireland) in Armagh was a Roman Catholic cathedral, but was in the hands of the Church of Ireland when this sort of switching ended. A much newer (Victorian) St Patrick's Cathedral in Armagh is the Seat of the Roman Catholic Primate in Ireland. This is just one of numerous examples of that sort of thing.
In 1979, when Pope John Paul II visited Ireland, he did not go to Armagh, because Armagh is in Northern Ireland, and at that time it was not politically correct for him to do so. Cardinal Tomás Ó Fiaich met the Pope and travelled with him in the Republic of Ireland only.