Only two of the defendants were in the dock on Thursday when the trial opened.
They were French former lieutenant Médéric Bertaud, 33, and Romanian former corporal Petru Sabin Suciu, 31, whose adopted name in the Legion was Adrian Streanu.
Both have left and found other jobs since the incident.
International arrest warrants have been issued for Chilean former sergeant Omar Andrès Martinez and Mexican corporal Wigberto Hernandez Canceco.
The four are accused of "wilful violence" leading to the death of 25-year-old Slovak Joszef Tvarusko, know as Matus Talas in the Legion, on an exercise in 2008.
Tvarusko was seen as a weak link by several of his comrades and, when the regiment was sent into the desert in 38°C heat and he complained of pains in his knee, he was hit, his water bottle was overturned and he was ordered to stay in the sun during rests.
When the group was climbing a 70° slope he collapsed and his body temperature reached 43°C.
The defendants' lawyers say their clients may have misjudged the situation and even made errors as commanders but do not deserve to be sent to prison.
Sentence is to be passed on 25 September.
The 6,800-strong Foreign Legion is almost entirely made up of foreigners, who can claim French nationality after serving five years or beforehand if they are wounded.