I would store anything that is unique to that user in the session and will be needed multiple times. For myself that is user name, permission level, and usually a unique identifier for the users account.
Reasons are, the name you may want to have it displayed in every page they load. The permission level they are set at can be used to hide or show links and/or give or deny them access to some pages. And the record number as that could be need if they wish to change personal info or if you want log that users activity as they visit different pages.
I guess my answer is a both and. Anything that is going to be needed more than just once as the user goes to each page on your site may be best kept in the $_SESSION. Otherwise if it a one time thing have the unique identifier grab it when or if it is needed.