i have read regarding this question of yours that the Tradition in Yemen was that in every family, on the Friday night preceeding the Sabbath, the father would teach his sons the weekly Parashath with te'amim, and have them read it back to them, teaching them all the melody and the wording. So that way when the child was of Bar Miswah age, he was already proficient with every Parashath and could read without need of a special teacher (although some were still hired if the family wanted to.) And this meant that every person called up was already proficient with every Parashath since he had been reading them with te'amim since childhood. i plan on attempting this custom. Check back with me in 20 years and i'll let you know how it worked out.
For a more detailed accounting, Rabbi Jacob Saphir published his travels to Yemen in 1866 in the book אבן ספיר. Here is an excerpt as given by the blog "On the Main Line":
He began this section noting that the Yemenite Jews all know how to read the Torah most correctly, with its proper syllables and accents and cantillation. He relates that up to the present the holy custom or earlier times, which is really the law in its most proper manner, is that all who are called to the Torah read it themselves - if one cannot read then he is not called to the Torah. For this reason the first thing they teach their small children is to read the Torah in this way, to the point where they literally have it memorized. In addition, they still have the venerable custom of old of publicly translating the Torah as it is read. They employ a 9 or 10 year old boy for this purpose, who translates each verse as it is read from the Torah. They do this also for the haftarah, which is read with a most beautiful tune. He also notices that they are extremely exacting in their reading, whether it is Bible, Targum or their studies in other books - in all these each syllable and accent and jot are read beautifully, as in [what Saphir imagines/ assumes] days of old. At one point Saphir relates something that amazed him, but was apparently perfectly normal. One shabbat (or should I say shabboth?) there was a want of a manuscript to read the haftarah. So the 18 year old cobbler who was serving him recited it (from 2 Kings 7) perfectly by heart.
However, in modern times, Yemenites come in all shapes and sizes and are often separated from their Mesorah. There is a Yemenite man at my synagogue who was raised by his Ashkenazi mother, and so he no longer has a Yemenite pronunciation, and sings with Ashkenazi melody.