I’ve worked on JSON API for a iPhone app development for a WordPress site to post photos but not using JSON API Plugin,
But the basic procedure would be: As you need to be logged in to publish your post, the communication is going to be two way
1. Author writes a article as draft on site A.
2. There is some link or button to initiate a JSON request to login to site B, you get a token if log in was successful.
3. You send another JSON request if you get token, with token and post data to site B.
4. If all the data is valid , site B publishes the Post and return you the POST ID
5. You can store the post ids for the record.
This would require you to check proper security. Also site b can’t automatically pull a new post from site a as you said in your first point, because something needs to tell it when to pull the info.
There might be a better approach I don’t know but you’ll require to make a custom plugin for that.