To copy and rename files, simply select the files, click on one of the selected files and drag them to the file list on the other side (other file panel).  When the file(s) are dropped, a popup menu will appear.  Simply select "Copy and Rename" from the menu.  You will be prompted for the new name(s).  A warning will be displayed prior to overwriting any existing files.

Access: Select files in one session.  Then drag and drop on top of the files in the other session.  Menu appears when the files are dropped.