The two face charges of defamation, which is a criminal charge in Thailand, and with offences under the Computer Crimes Act.
If found guilty they could face up to two years in prison for defamation and five years for breaching the Computer Crimes Act, as well as a B100,000 fine.
The charges were formally brought to trial after a complaint filed by the Royal Thai Navy. The complaint relates to an article published by on the Phuketwan website in July last year.
The article quoted an investigation by the Reuters news agency, which alleged that some members of the military were involved in trafficking Muslim Rohingya refugees who had fled Myanmar.
Testimony will be heard today through Thursday, before recess. A date for the trial to resume has yet to be set.
The case has drawn criticism from human rights and press freedom groups around the world, with many international observers and members of the press present at the Phuket Provincial Court today.