The Fourth Amendment prohibits "unreasonable searches and seizures," which the Supreme Court has broadly interpreted to include things such as warrantless wiretaps [1] and even certain types of cell phone call metadata [2]. Effectively, if a person expects their conversations to be private, and society views these as reasonably private (e.g. not taking place in the open, where anyone could overhear), warrantless covert government access constitutes unreasonable search and violates constitutional rights. To quote Supreme Court Justice John Marshall II:
"My understanding of the rule that has emerged from prior decisions is that there is a twofold requirement, first that a person have exhibited an actual (subjective) expectation of privacy and, second, that the expectation be one that society is prepared to recognized as "reasonable." Thus a man's home is, for most purposes, a place where he expects privacy, but objects, activities, or statements that he exposes to the "plain view" of outsiders are not "protected" because no intention to keep them to himself has been exhibited. On the other hand, conversations in the open would not be protected against being overheard, for the expectation of privacy under the circumstances would be unreasonable."
Note that this statement was in agreement to a court decision saying that government wiretapping of a public phone booth was unconstitutional, because when you're in a phone booth with the door closed, you don't expect people to overhear you, and thus it's an unreasonable search and a violation of constitutional rights for the government to record your conversation. Note that this doesn't depend on notions of "trespass" or specific areas, and rather depends on a person's expectation of privacy:
"For the Fourth Amendment protects people, not places. What a person knowingly exposes to the public, even in his own home or office, is not a subject of Fourth Amendment protection. But what he seeks to preserve as private, even in an area accessible to the public, may be constitutionally protected."
OP is almost certainly correct that dragnet, warrantless searching of private message data by the government is a violation of constitutional rights, although it'll take a Supreme Court case to make a real ruling on the subject.
[1] Katz v. the United States (1967) https://en.wikipedia.org/wiki/Katz_v._United_States
[2] Carpenter v. the United States (2018) https://en.wikipedia.org/wiki/Carpenter_v._United_States