After doing some digging around this, we discovered that BigSur doesn’t let Local automatically trust SSL certificates. They should still be registered with the system, but you’ll need to manually search for and trust each Local site’s certificate as outlined in this help doc:
https://localwp.com/help-docs/ssl/managing-a-local-sites-ssl-certificate-in-macos/
We’re exploring alternative ways of making this easier, but in the meantime, you’ll need to work through the steps outlined in the help doc.