Everyone has that access. Click on the padlock icon on any HTTPS-using website, and after a few more clicks you can export a copy of the site's certificate (at least on Firefox).
But that gains you nothing without the corresponding private key. The private key is generated on the website's server, and is never sent to the certificate authority (what is sent is a "certificate request", which has basically the same information found on the certificate).
> are there ways for me to know when my own browsing is truly private between myself and only the party at the other end, and not other curious or intrusive uninvited third parties?
Now that's a different question. While having access to the certificates is no problem at all, being able to create a new certificate for an arbitrary website allows one to pretend to be that website. The only defense against it is that, if a CA is caught issuing these certificates, it risks being removed from the browser's trust lists, which is a death penalty for a CA's business. Also, there is a new initiative (Certificate Transparency) to make it easier for these certificates to be caught.