There are some people who wrongly share private information. There are others who (usually reporters) who pester the sick and abused. Still others bring their conflicts into the public. All of these things are wrong sometimes very wrong. One possible solution to these problems is to restrict the press (or individuals generally) from releasing this kind of information. This counts as a restriction on free speech because one is restricted from stating opinions. Such a restriction is not worth it the costs are much greater than the benefits.
We must remember that every wrong act cannot be criminalized or punished in civil courts. Shunning someone because you dont like them, name-calling and watching someone drown are all wrong. But none of those acts are currently punished. Nor can they be punished. In order to punish any act that is wrong, the cost of punishment must be outweighed by the benefit of punishment.
Reporters who harass individuals, trespass, break and enter, lie or threaten break laws. As long as these laws are enforced quickly and effectively, much of the wrong found in the above actions is already punished. Courts are slow, ineffective and favor the rich. But that is problem with the courts, not a problem with the law. The solution is fix the courts, not to restrict free speech.
The cost of adding a restriction to speech depends on how the restriction is added. Those who have the money will sue, and their sensitive informationwill never reach the public. This is especially true if they are both rich and criminal. Those who do not have money will still have their sensitive information leaked, especially if they offend someone powerful. If reporters are restricted, this will cause a chilling effect in the depth and informative nature of their reporting. If individuals are restricted, then this will increase radicalization as individuals will share sensitive information only with their friends and those they trust not to rat them out.
The benefit is minor. When we restrict ourselves only to those cases in which no other laws are broken, this prevents emotional harm. I grant that emotional harm can be a big deal. But it is rather small when compared to the harms of radicalization, protection of rich criminals, and a chilling effect on free speech generally.
No law should ever be enacted when the costs are greater or equal to the benefits. In this case, the costs are much greater than the benefits. Instead of restricting the free speech of others, why dont we freely and voluntarily restrict our own speech and avoid gossip.