Each church can recognize it’s own saints, so you will find some saints that are venerated in just one church. In both East and West, it starts out with popular devotion to a particular (holy) person. If there is a big enough crowd with a devotion to the same person, the person might just be recognized as a saint by his or her own church. The Eastern churches don’t really have an elaborate process for sainthood like the Latin church does. When the Latin church canonizes someone as a saint, the Eastern churches (in communion with Rome) usually recognize it as well out of courtesy as part of the universal Church (but not always).