drupal 9 tempstore.private | user.private_tempstore VS tempstore.private
in D9 user.private_tempstore is deprecated in favor of tempstore.private.
I wondered if this meant that things kept in tempstore could exist between user sessions. This infact is not the case though.
Per the class explanation on Drupal.org:
"The PrivateTempStore differs from the SharedTempStore in that all keys are ensured to be unique for a particular user and users can never share data."