Courts exist for this reason. On one hand, you have the right to privacy (though it is not spelled out that clearly, it is inferred).

On the other, you have freedom of the press clearly outlined in the first amendment. There is no perfect answer and it will be subject to some debate. But in this particular instance, precedence has made it clear that people of significant stature (and that could mean a blue check is a signal of that in court!) have less of a right to privacy when weighed against the first amendment.