If you are in CPM 713+ and Interaction Portal, the best practice is to create requestor level data page to share the data.This way, you can only maintain single copy of data. But if you have requirement to keep a copy in service case, there is a data transfrom in PegaCA-Work.pyDefault (or something in this hirerachy) which will be exeucted for every case creation. you can override this DT to copy of data from D_Interaction Page.
If your requirement is to use/copy a named page created in Interaction thread into service case page, temporarily you can keep this page one of the available requestor level pages in a page property and copy it when a service case is launched in pyDefault/NewDefaults/PerformDefaults activity.
If you are using CSR Portal portal, you can promote pages to threadfamily scope which will be available for interaction and service case threads.
please let us know if your requirement is something different.