It has been argued that blackmail should be legal if gossip is legal, and even that there are no good consequentialist counterarguments (!). I think this isn’t obvious because the disclosures incentivized by blackmail are systematically worse than gossip.
Gossip has costs and benefits: on the one hand people learn information they can use to make better decisions (e.g. who they want to associate with or trust), but on the other hand spiteful gossip hurts people and so can be costly for the same reasons as violence. (See the appendix for more discussion of spiteful gossip.)
Outside of blackmail, people have more reasons to gossip when the information is actually useful to the listener, whether out of kindness or reciprocity. In contrast, when blackmailing, people’s only motive is to share information based on how much it hurts the subject. So talk motivated by blackmail is expected to have fewer benefits, and more harms, than the average gossip. So “ban blackmail but not gossip” looks like a reasonable position.
That’s a simple argument, but it’s enough to convince me that “ban blackmail but not gossip” is a reasonable position, and I haven’t seen it rebutted. (I do think people’s opposition to blackmail seems irrationally strong, but I myself am tentatively opposed except for blackmail about breaking the law.)
Appendix: “spiteful” gossip
By “spiteful” gossip I mean gossip motivated by the desire to cause harm. Most gossip will have a complex mix of motivations and be spiteful to some intermediate degree. Gossip prevented by a blackmail ban is all spite.
If we could ban spiteful gossip, that might be even better than just banning blackmail, but the distinction seems impossible to litigate (since useful gossip also frequently involves saying bad things about people).
Is spiteful gossip zero-sum? Some have argued that even spiteful gossip isn’t so bad, because when people think worse of one person they think a little bit better of everyone else by comparison. But even if any given instance of spiteful gossip helps others as much as it harms the target, everyone will work inefficiently hard to avoid being the subject of spiteful gossip.
Not only is this an Econ 101 objection, it seems to track the reasons people are opposed to legalizing blackmail (people often say they don’t want a world where everyone is constantly watching their backs).
Are the incentives good? You might hope that’s not so bad either, because “avoid being the subject of spiteful gossip” might be aligned with “behave prosocially.”
But there is a weak correlation between “costs of others knowing X” and “efficient disincentives for X.” (The costs are often determined by signaling effects or by equilibria of adversarial games.) And even if these were aligned, spiteful gossip incentivizes concealing information in addition to actually behaving better, which is a major cost.
It’s possible that spiteful gossip on average does good by improving incentives, especially within particular domains. For example, I think blackmail about law-breaking probably has net benefits, because (i) the costs of “others knowing I broke the law” are designed to be reasonable disincentives for breaking the law, (ii) most people don’t have too much to hide and so there aren’t big costs from “watching our backs.” But I’m suspicious in general, and there’s no way the question is obvious.
The sideways view 
“people’s only motive is to share information based on how much it hurts the subject”. Blackmailers can have many diverse motives, just as with most human behavior. Part of that can be a sense of justice, that people shouldn’t be able to get away with bad behavior. Blackmailing can punish bad behavior. Even if their motive is to gain money, that isn’t such a bad motive; much or our society is built on it. And they seek info that hurts would people but would help others, so it isn’t just about hurting some. And if you could identify a bad motive that mostly only gives bad behavior, you could bad gossip based on that bad motive as well.
Perhaps I should write “optimized” instead of “motivated.” I don’t really care about the distinction between seeking justice, punishing bad behavior, threatening, etc., they seem to have the same analysis, it’s about the distinction between “I’m saying it because it hurts someone” and “I’m saying it because it helps someone.” I’m fine with people trying to make money, but again it matters whether my action makes money because it helps someone or because it (in expectation) hurts someone.
(Incidentally, I don’t see why the justice or punishment motive is served by blackmail—presumably that motive is better served by just directly revealing the info. All the extra incentive effect from blackmail factors through hurting the subject of the blackmail.)
“And they seek info that hurts would people but would help others, so it isn’t just about hurting some.”
I don’t think that’s necessarily true. You can blackmail someone pretty effectively by threatening to send embarrassing photos of them to their family, but the family is not benefiting from seeing those photos.
A legal blackmail market creates a cash incentive to find and defend against blackmail. Finding and defending against blackmail has costs. Blackmailing does not add value. Therefore, prima facie, a legal blackmail market is a negative sum game.
Unless blackmail somehow not only adds value to society, but it adds so much value to society that it justifies the creation of negative sum games, I would not be a big fan, myself.