The English court of law known as the Star Chamber (1398-1641) is the primary example. It questioned witnesses and defendants secret. Inevitably, it became an abusive political weapon of the Crown. Our Founders were well aware of its notorious outcomes.
As Justice Hugo L. Black observed in the 1948 Oliver case:
“The traditional Anglo-American distrust for secret trials has been variously ascribed to the notorious use of this practice by the Spanish Inquisition, to the excesses of the English Court of Star Chamber, and to the French monarchy's abuse of the lettre de cachet. All of these institutions obviously symbolized a menace to liberty. In the hands of despotic groups each of them had become an instrument for the suppression of political and religious heresies in ruthless disregard of the right of an accused to a fair trial. Whatever other benefits the guarantee to an accused that his trial be conducted in public may confer upon our society, the guarantee has always been recognized as a safeguard against any attempt to employ our courts as instruments of persecution.”
As with many of your natural rights, judicial sophistry since 1791 has eroded the public-trial right, progressively finding it “not absolute”. Instances of courts somehow failing to maintain “order and decorum" resulted in partly closed trials. Minor victims in sex-offense trials were then protected. Then “sensitive information” of undercover witnesses started to be used to shut out the public.
By the late 20th century, a “state secrets privilege” found a way into legal precedent. Government simply began to assert sensitive information, if used as evidence, might endanger national security. Our supine judiciary almost never looked at the evidence to evaluate whether the Government can justify its claims. Abuse was inevitable.
In 2011, Thomas Andrews Drake, a former NSA executive, was persecuted for blowing the whistle on unconstitutional surveillance. Federal prosecutors made several moves to force Drake to suffer trial in secret, pressing the case under the Classified Information Procedures Act and "silent witness rule." Both anathema to the natural right of public trial.
Due to the unprecedented display of tyranny, the Government soon abandoned all felony charges against Drake. The presiding judge told the federal prosecutors they had been "unconscionable" to charge a defendant with a list of serious crimes that could have resulted in 35 years in prison, only to drop all of the major charges on the eve of trial. However, the robed coward did nothing about it. Drake had been financially devastated.

When witnesses cannot hear the testimony of other witnesses, lies are encouraged. When witnesses are secretly questioned, testimony that would shame them if made publicly is emboldened. When the accused are tried in secret, confidence in the criminal-justice system and its officers is lost. When shiftless judges forsake their responsibility to demand security clearances and be read onto Sensitive Compartmented Information and Special Access Programs, and an ignorant Congress allows the executive branch to deny them the same, judicial review vanishes. When public trials disappear, dishonorable agents of the state attack political enemies. When the judiciary is cornered into dismissing entire cases against the Government because trials cannot be open to the public, constitutional crisis ensues.