Hmmm...
If you're asking what I THINK you're asking, there are a couploe of misconceptions need to be cleared up. Firstly, the database runs ON the webserver where the site is hosted. Most databases will not accept external connections, but there is a way to do it.
What you could try is (provided your free host runs php, and you have permissions to use scripts) is create say a form. In turn this form would create a custom made php file, which would then be uploaded to the paid server, and excecuted. Then you need to tell the paid server to send the info back to the free server and display that information as the result of the form....not exactly easy hehe 🙂
ALternatively, copy the whole database ontop the free host? You will need to check if you have permission to use php/mysql on the free server, but that would be the easier way...
Hope that helped,
RM.