The bottom line is that government NDAs to protect classified information and private trade secrets are common and legitimate. But NDAs aimed at muzzling people from speaking about anything relating to Donald Trump are not legally authorized, and in fact probably violate both whistleblower laws and the First Amendment.
Still, even if the NDAs are not legally enforceable, the fear of retribution from Team Trump is enough to keep people quiet—which is why Congress must step up and pass a new law that bans this practice outright, with a private right of action to sue the government for money damages and injunctive relief if a future administration were to pick up the Trumpian baton in this regard.
The Constitution does not contain any language requiring transparency in government. Congress has passed a number of statutes ensuring that government actions see the light of day, most prominently the Freedom of Information Act (FOIA), which affords regular people—and the press—the ability to petition the federal government for disclosure of records. (To the extent that private citizens like Rudy Giuliani do the work of the federal government by, say, acting as faux diplomats on personal missions for Trump, their records are not covered by the FOIA.)