The others called were at least the twelve named in 3 Nephi 19:4. Of course, the Savior could have called more to have the authority to baptize, but for the sake of symmetry with his Old World ministry, it seems reasonable to conclude that he called just these twelve, who could then confer authority to baptize on others.
Apparently the mode of baptism had caused some disputation (see also v. 28) since the Messiah states he that he will instruct them correctly. This problem has never before been mentioned, which simply indicates how incomplete Mormon’s picture is of the Nephites. There are certainly many more areas of which we are similarly unaware because of Mormon’s editorial choices.
This text on the mode of baptism is placed between the Messiah’s appearance to the Nephites and his Sermon at the Temple discourse (paralleling his Old World Sermon on the Mount). Both events are clearly meant for the whole group, so it seems unusual for a session to occur between the two that was obviously communicated only to the twelve disciples. I hypothesize that its placement may be due to Nephi’s reconstruction rather than to its literal sequence. As Nephi constructed this account perhaps a month or more after the fact, baptism as a sign of accepting the Messiah’s mission was so important that he communicated its primacy by moving it earlier in the sequence of events. Nephi is writing a religious history, not a chronicle. Spiritual matters were much more important than temporal matters, and certainly more important than the exact sequence of events.