How does the Catholic Church handle documents worldwide nowadays?

Marriages and baptisms certainly generate certificates that the church needs to store somewhere. Catholic schools such as the Marist Brothers and La Salle Brothers, with hundreds of thousands of students worldwide, also most probably report information back to the Catholic Church. I was wondering how the church handles all this information. Are they stored locally, regionally, nationally or does the Vatican receive everything back? Can someone in the Vatican in higher positions inside the church easily look up if a certain someone born in another continent was baptised or got married? Or maybe if someone studied in a Catholic school? And how much of the church's bureaucracy is digital nowadays?