A sound and balanced answer requires some further clarification in the original question.
What is the status of the complainant?
If a Cllr, then they may be subject to adopted policies of the council in relation to dealing with staff complaints and as such that ‘policy’ might require a ‘process’ of investigation which ‘could’ be subject to confidentiality until the finding of an internal investigation is completed
somewhat bizarre but whilst no individual Cllr may direct the clerk, all individual Cllrs (as individuals) are held as being the ‘employer’ of the clerk. Consequently, if a Cllr complains about a clerk, they should submit that complaint to the HR/staffing committee and let that committee run its course. A clerk ‘could’ counter a valid complaint by seeking to invoke ‘employer bullying’ by an individual Cllr if an individual deviates from corporate policy.
Conversely, no such restriction automatically applies to a member of public.
It ‘could be’ that an individual raising a complaint does not receive a satisfactory response from a PC investigation into clerk behaviour (not at all uncommon.)
A member of public is at liberty to discuss, publish, advertise anything they so please within the existing bounds of freedom of speech.
The potential for a civil claim for defamation is vanishingly unlikely since there are several defences against such an action: truth, honestly held belief, public interest etc.
It is also prohibitively expensive (and NOT appropriate for use of tax payers money by a PC on behalf of an employee nor a Cllr to advance a civil claim.)
So, a Cllr should (probably) allow proper process (if they have confidence in it) which would (probably) reasonably require non disclosure prior to completion but neither a Cllr NOR a member of public is legally obliged to keep schtum if they don’t want to.
Perhaps Kier Stalin’s thought police might arrest them for walking the wrong way around the wheel though…