Hello,

it depends on what you have done on the local version for the mobile layout. If you worked mainly on css, you can copy the local css files of Responsivizer to the remote host. If you have created some modules for mobile the database table of Joomla that holds modules is involved.
About the settings of Responsivizer, they are stored in the #__extensions table of Joomla.
If you need to transfer all setup at once an interesting solution would be using Akeeba backup to clone the site on the remote host.

Best regards

John