Ex Wikipedia: Over 186 constitutions mention the right to privacy
https://www.constituteproject.org/constitutions?key=privacy
Oh, and sniper: because the right to privacy will be defended by a number of individuals with radical violence, as you should know if you ever visited the world, so evidently I should not even need to proceed from «Yes. Implicitly».
So contextually: if it were not anonymous, it would not be used (but by a specific class of entities).
> The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized
In legal matters, those two rarely converge. On technical matters both are usually abstracted as "keeping information secret", but even then they often require widely different techniques to be preserved.
But the context here can hardly be instanced in the case of "conversation". Before protecting exchanges (in privacy and anonymity), there is the act of accessing information that would make an intruder an eavesdropping peeping tom.
It is more clear in its actual context: the "private space", which could be breached by so-called "intrusion upon seclusion" cases (peeping, eavesdropping etc.), is protected - see e.g. the case of the "IV amendment" (discussed nearby), protecting against the intrusion in one's «papers» (instance of a broader private space relevant to the explicit context of «searching» as primary practical concern), and base for more specific legislation, possibly after Warren and Brandeis' "The Right to Privacy" - which is relevant because further legislation got needed after new technical-aided phenomena (the gossip press) breaching base principles otherwise formerly threatened by «the idle and... the vicious» (Warren and Brandeis).
So: having rights about a private space, one's act of accessing information must be in anonymous form.
Edited to Add: All bets are off now, as the court has explicitly reversed stance on this as it regards abortion, which leaves a big precedent for expanding future privacy “exceptions”.
They are all cases of instances of breach of a basic principle - license of intrusion is limited, naturally -, which in law needs to be made explicit to sanction specific cases, but remains otherwise evident (in the case of privacy, its breaches are easily farcical).
In Europe, the European Convention on Human Rights (1950), signed by 46 States, specifies in Article 8: «Everyone has the right to respect for his private and family life, his home and his correspondence» (again, an example picked for prominence).
Please also note with reference to your «All bets are off now» that discussion will be made about the boundaries of the principle in practice - the Principle remains.