Are COR's allowed to distribute copies of the PWS to persons who don't have the need to know?
This is a really good question! You were essentially correct in your question background. To answer your question; no, you are not authorized to distribute copies (even to people who have a need to know) unless the Contracting Officer gives you specific instructions to do so. Even then, if I were the COR I would ask the contracting officer to accomplish that.
I highly encourage you to discuss this situation with your contracting officer as soon as possible. Especially (emphasis added) if these requests are coming from people outside the government (other contractors and even the contractor you are dealing with) as this could be a FOIA issue.
The rationale is because the Federal Acquisition Regulations, Defense Federal Acquisition Regulation Supplement, and the Air Force Federal Acquisition Regulation Supplement (PGI) all have policy, procedures and guidance on how contracts shall be distributed.
I do not expect CORs to know how to navigate the federal acquisition regulations or supplements. But if you need some references to help facilitate initiating a conversation with your contracting officer; here they are:
FAR subart 4.2 Contract Distribution, DFARS subpart 204.2 (and the associated PGI), and AFFARS PGI 5304. FAR subpart 24.2 Freedom of Information Act, and the DFARS/AFFARS corresponding citations.