How does a rabbi choose his disciples?

If this question is referring to the Modern Age, it is rather difficult to answer. Judaism is not monolithic in the way that Catholicism is monolithic with one council that renders all decisions for all Catholics everywhere in the world. Judaism is congregational and its congregations array themselves in accordance with the tenants of different movements. These movements have institutions of higher learning such as Hebrew Union College for Reform Judaism, the Jewish Theological Seminary for Conservative Judaism, and hundreds of Yeshivot for the different strains of Orthodox Judaism. Jewish leaders in their respective movements go to these institutions to become versed in the interpretations of their chosen sect and then apply to jobs as rabbis and hazzanim in various communities. If the rabbi or hazzan is seen as an exemplar of the movement, they may be asked to attend meetings that decide the future decisions of that movement.