The Talmud clarifies the meaning of the statement that opens the mishna.

Gemara: At the beginning of the discussion, the Gemara assumes that “one” means an entire tribunal, composed of three judges. It therefore asks: What is the meaning of: This litigant chooses one for himself and that litigant chooses one for himself? Isn’t it sufficient to adjudicate the case with three judges? There is no need to select two entire courts, a total of six judges, to choose three other judges.