MEXICO CITY—When a top Mexican or Colombian drug lord is captured, events normally go something like this: He gets extradited to the U.S. and makes a closed-door deal with prosecutors to give information on the drugs trade while getting a reduced sentence in return. The public finds out little to nothing of the details.

But the upcoming Chicago trial of the son of one of Mexico's top drug lords has broken all the rules. This time, Jesús Zambada Niebla is going mano a mano with U.S. prosecutors, with both sides trading allegations...