The DMCA takedown notice procedure is governed by 17 USC 512 subsections (c), (d), (f), and (g). Under 17 USC 512 (g)(2)(A) the service provided is expected to notify the subscriber who places allegedly in fringing content on the online service when it is taken down due to a notice. Under 17 USC 512 (g)(2)(B) the contents of a counter notice may be provided to the person who filed the original takedown notice.
There is no other provision under 17 USC 512 that either forbids or requires publicizing either the existence or content of a notice or counter-notice.
Note that the DMCA is a specifically US law and procedure. A number of other countries have somewhat similar procedures governed by their own law, which may be loosely refereed to as DMCA notices, but are not in fact under US law. Some providers not subject to US law choose to accept and act on notices formatted to the DMCA standard, although they gain no legal advantage from doing so.
Note also that even within the US, no provider is required to act on a DMCA notice or counter-notice. But if such a provider does not honor notices, the provider loses the safe-harbor protections of 17 USC 512, and may have greater liability if a court case follows.
If a provider is subject to the GDPR, or the CCPA, disclosing notice info beyond what is permitted or required by law might be a violation of such a privacy law, but neither of these laws currently permit a private individual to file a lawsuit for such an improper disclosure.