I’m sorry for leaving this topic dangling, but I was mostly at a loss for what might be causing this.
I think that this is a related issue to another topic where not including the trailing slash under apache causes the issue:
Maybe neither ? 
I think that what’s happening is that because Local’s router (which is basically a proxy) is running nginx, but the end site is running Apache, there’s something incorrect with how the two services are communicating.
In the above reply I linked to, I recorded my QA session and at around the 2:00 min mark and the 5:20 min mark, I get odd HTTP response codes for the various URLs. For example, as a logged out user, trying to access the wp-admin, I think there should be a 302 response to the wp-login.php page but instead there are 200 responses.
For the Apache site, it just breaks but for some reason nginx fails gracefully and shows the site.
I’ve filed a bug with the Local team, but I don’t have an ETA for when it will be worked.