It could be some sort of setup with that Mac. Is this a work computer, for example, did the company implement any sort of security settings – I’m thinking of something like issuing their own SSL certificate or CA authority.
One other idea – How long have you had Local installed?
I wonder if maybe there’s an old version of the PHP service that needs to be updated? You could try first backing up your sites, and then the steps outlined in this reply, which deletes the current PHP service and downloads the latest ones: