After browsing incognito for a while, I was able to reproduce the cookie notice issue on specific pages on our site, where accepting while viewing that page would have no effect (but if it was previously accepted on a working page, then the cookie notice would not be visible on the non-working page). There appear to be significantly more pages that work than don't, but the forums index was one of the non-working pages, and it is a common landing page.
This issue is now fixed on our site.
However, I still have not had an issue with the Privacy Policy notice. In fixing the cookie notice, the CDN is forced to re-pull the script that generates the notices. It is possible that there was a caching issue with the CDN and this was causing an issue with the notices. Having the CDN re-pull the scripts should fix the Privacy notice if that was what was causing it.