In a shared site, how to get products in a category from one site to another?

Since you’re not currently running a multi-site install, you can either install the WP-REST-API plugin which is currently in beta, or you can wait for it to hit core which will probably be around version 4.3 or 4.4.

If you were on a multi-site install, you could have easily used the switch_to_blog() function to switch between blog databases to query databases.

WP Rest API Docs : http://wp-api.org/

Evan