Two Database Instances, need to share table data

**Background** Two sites, both WordPress; one is public-facing, and the second is private with restrictive access. The public-facing site has a form on it, but I need to be able to access the information received from that form so that the private site can review it. The form is built through a WordPress plugin. Both databases live on the same server but are different instances and are managed through phpmyadmin. Once the business logic is handled on the form, I don't need to retain the information on the public site and prefer that it get deleted from the table. **Question** Does anyone have a recommendation on how I can share information/transfer information to the private site database without losing functionality on either site?