The side that wins the challenge will receive a message via the Character > Messages screen from the leader of their faction, this message has a thank you note and also an attached item.
The item is a badge that can be taken from the message and then used from your Inventory screen, once used the badge is placed into your badge/special trophy list and will show up on your File > Trophies Earned list.
From that screen you can use the badge to claim special rewards such as new titles and any other rewards that are added at a future date.
Remember one of the main rewards for taking part in factions is the faction fame which earns you rewards from your faction leader while the challenge is running.