In a statement, the NOHRS said: "They rejected all accusations made against them and stressed that the Damascus Declaration was not an undertaking [carried out] in a secret manner and that its aim was to defend the homeland."
It is the biggest collective trial of Syrian dissidents since 2001 after the so-called Damascus Spring, the brief period of relative freedom of expression that followed Bashar al-Assad's, Syria's president, rise to power in 2000.
The court adjourned the case until August 26.
Among those being tried are Fidaa Horani, president of the NCDD, Ali Abdallah, a writer, Walid Bunni, a doctor, Akram Bunni, a writer and secretary of the NCDD, and Riad Seif, a former member of Syria's parliment.
The 12 are charged with "spreading false information which weakens the morale of the nation and national sentiment, joining a secret organisation with the aim of modifying the nation's political and economic status, incitng racial and sectarian dissent and harming the state," according to Syrian non-governmental organisations.
Several lawyers and Western diplomats attended the trial opening.
The NOHRS called on the authorities "immediately to release all detainees" in Syria and "halt proceedings against them so as to consolidate national unity and preserve citizens' rights to free speech guaranteed by Syria's constitution and the international human rights charter".
Another human rights group, the Syrian League for the Defence of Human Rights, denounced the trial of political activists and also called for their immediate release.