As expected, seasonal characters can be played offline on console (save data is local). A connection to PSN or Xbox Live is only needed during character creation and when starting/continuing seasonal play. Seasonal journey/leaderboard progress is only updated with an active connection to each consoles respective network. After that, an active network connection is no longer required.
Blizzard has implemented some changes to reduce the chances of duped items including disabling all trading in network multiplayer and couch co-op. Items that are dropped on the ground cannot be seen by others. Attempting to do the same in a game hosted by someone else also doesn’t work as restoring an older save and reconnecting back to that hosted game will cause those drops to disappear. The mailbox is also disabled in terms of being able to send items.
These restrictions are an unsurprising but very fair trade off to allowing seasonal play without making large changes to the way the game works on console (local binary engine).
