InfoPath form exported from Sharepoint list, how to publish to another SP list?

 Hello,

I exported a list from SP list into InfoPath. It works well and it has nice formatting. However I would like to publish the form into another SP list while keeping the data into the original SP list. This because I would like to have the form accessible to everyone, but access to the collected data should be controlled.

The only option available under 'publish' is the SP list where the source/destination data is.

Any suggestion appreciated. Thanks.