For the first Christian centuries nobody made saints, because there was no procedure to make them with. A martyr was buried, the community kept the anniversary, and the bishop either permitted the gathering or did not. Peter Brown's account of the late antique cult describes a practice organised around graves and around the patronage a dead friend of God was believed to exercise for the living, and it grew from below rather than being conferred from above. Bishops regulated what was already happening in their own dioceses, and cults spread by travel, by relics moving, and by imitation.
The shift to a papal decision was slow and never as clean as later summaries suggest. The canonisation of Ulrich of Augsburg by John XV in 993 is the case usually cited as the first performed by a pope, but it did not stop anybody else performing them. In 1170 Alexander III wrote to a Swedish king that no one was to be venerated as a saint without the authority of the Roman church, and that letter entered the collection of decretals Gregory IX promulgated in 1234, which is the point at which reservation to the papacy became ordinary law. Andre Vauchez's study of the medieval causes shows what changed in practice: the inquiry acquired procedure, witnesses, notaries and a written record, and the question moved from whether a cult existed to whether the life and the miracles could be proved.
The machinery has been rebuilt repeatedly since. Sixtus V created the Congregation of Rites in 1588 to run it, the 1917 code of canon law absorbed it, and Paul VI separated the causes into their own congregation in 1969.