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).
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.