A very good question and issue!
Historically, whenever churches separated from the Catholic Church either by reason of schism, heresy or both and then reunited with the Catholic Church, it was often the case that any new saints that were venerated by those churches/communities continued to be venerated.
The majority of saints were and are local saints, so this did not pose a problem. According to Fr. Holweck in his “Dictionary of the Saints” 1924, Arian saints are in the Catholic calendar including St Artemius the Dux Augustalis of Egypt who was killed while he destroyed pagan temples. St Nicetas the Goth and St Sabas the Goth were Arian or Semi-Arian priests. Fr. Holweck gives other examples of saints from other heretical communities etc. Their martyrdom for Christ allowed the Church to “overlook the defect of their orthodoxy.”
Some of these saints have, over time, made it into the Roman canon, in the same way that Orthodox-canonized saints have via Pope Pius XII and Pope Paul VI listings.
When Orthodox Churches entered into communion with Rome at various times in the past, their Orthodox saints and miracle-working icons continued to be venerated. Russian Greek-Catholics venerated and continue to venerate all the saints on the Russian Orthodox calendar (e.g. St Michael’s Russian Catholic Church).
In all cases, canonizations/glorifications were conducted by the local Church. Even with Pope Urban VIII’s decree that beatification was reserved to Rome alone, many Roman Catholic bishops continued to beatify their local worthies and establish feastdays for them. For example, Bl. John Duns Scotus the Franciscan theologian who outlined the doctrine of the Immaculate Conception in his day was beatified locally by an Italian bishop and was venerated as such for a very long time before Bl. Pope John Paul beatified him in recent years.
Although St Thomas More was only canonized in 1935, he was, in fact, venerated as a local saint in Rome since 1575 or forty years after his martyrdom. The same is true of St John Fisher. Peter Lombard was and still is locally venerated in several regions of Italy as a local Beatus (and I don’t believe Rome has approved his cult, but it continues nevertheless).
Jerome Savonarola the Dominican firebrand preacher and moral reformer (also an opponent of Pope Alexander VI) was highly venerated within the Dominican Order and in the city of Florence as the “Apostle of Florence and Martyr etc.” Medals were struck of him and, in the 15th and 16th centuries a special Mass for “Il Beato Girolamo Savonarola” was celebrated by Dominicans. Of course, this was all done without Rome’s approval and even in the face of Rome’s DISAPPROVAL.
Savonarola had died excommunicated but the society that promotes his canonization today have uncovered a papal document by none other than Alexander VI where the excommunication was dropped just prior to the Dominican’s death. The Jesuits are very much against Savonarola’s canonization because he opposed Alexander VI.
Pope Julius II wanted to canonize Savonarola but was told that the Brogias might not look favourably upon that . . . There is a portrait of Savonarola in the Vatican, but without a halo. The Archbishop of Florence has re-introduced his Cause. St Philip Neri and St Catherine of Bologna were both from Florence and were both personally devoted to Savonarola. When St Philip Neri died, a medal of Savonarola was found around his neck. This was reported to Rome during his canonization process as a possible impediment, but Rome ordered it to be “passed over” or “praetermissus.”
In the Orthodox East, canonizations were likewise done by local monasteries as they continue to be done on Mt Athos and elsewhere today. The more than 150 + Venerable Saints of the Kievan Caves Lavra were at first only locally venerated at the Lavra, but then St Peter Mohyla of Kiev canonized them in the 17th century for church-wide veneration. (the second part follows . . . sorry for being so long-winded . . .)
Alex