I personally am waiting for the "data breach of the data breach response" news to hit - I'm sure it will some day :)
The first question is one of trust. Why would one trust companies, which are often hand-picked by companies that could not protect your data in the first place, to have access to more of your data than they would otherwise have (as with continual credit monitoring). I'd give high odds that many of these companies have, deeply embedded in their terms of service or some other document no one bothers to read, authorization to package and sell your data (or aggregate data including yours) to other firms as an additional way to attain profit. And if not, they wouldn't be the first to go ahead and do it anyway.
As a case in point for why automatically trusting "identity protection" firms of any kind is a bad idea, one need only look at the history of LifeLock and it's $100-million+ fines by the FTC in 2015 for failure to protect user information and engaging in deceptive advertising (Symantec bought them for $2-billion plus shortly after that, because of course). In short: the very company promising to protect people from identity theft was negligent in protecting data that could cause people to have their identity stolen. The top equity holders still got massive rewards for it, because the lies were convincing and the fines were small in comparison to the profits derived from them.
So thus my advice is: who exactly is offering the credit monitoring? If they can monitor your information "for you", then they are also entitled to a number of liberties in retaining, reporting, and otherwise using that data for their own purposes as well. Do they have any sort of track record that you would trust in doing their jobs and not using the misfortune of others to drive up corporate/personal profits?
On the flip side, maybe if we sign up for every free credit monitoring offering we get that results from a data breach, we could get into more data breaches, which will result in more free credit monitoring, until we have every credit monitoring service in existence monitoring our credit for us! Which would be a great trick, if only their services were likely to be worth anything at all...
By all means, make sure you have a way to check up on your credit regularly, but I am aware of no evidence of any kind that suggests that exposing your data to additional companies is likely to make you any safer. None of them seem to offer anything with teeth in the way of actual protection, just more data sharing. You'll certainly be more monitored by more people, but at some point that becomes a bad thing.
