Two live WP websites – how to sync?

You have basically tree ways.

  1. WP API (JSON)
  2. Feed (XML)
  3. XMLRPC API

I think the first solution via WP JSON API should be your prioritized way. The WP API will implement inside the core of WordPress in the next releases. Currently you can use it via plugin. I think, this is the standard for the next time for work with data from the WP install. Also you find a lot of tutorials, examples to use the data.
For the authentification use OAuth, like this plugin. You find much more hints, tutorials in the documentation about the API. Also interest is the actually hint from the developer side.