'Priced Possession' was a term used for the item stolen from the opponent upon victory in a PVP, and with the notification, options whether to be claimed the stolen reward or to register would become available. If claimed, the reward becomes just another artefact(ed) tool to wield, but upon registering, the reward will be forsaken when defeated in a PVP match, and the chances were 100%.
If so, why register?
Hmm....commonly, for two reasons~
One, the higher the registered weapon's rank, the more, the greedy, will challenge you to posses the registered weapon.
Two, because for people like Ken, who craved for exciting and blood boiling fights, artefact(ed) weapons weren't all that useful since they only made fights easier and boring.
Who'd want to one shot their enemies? And, where was the fun in that?
And, yes, the registered possessions is visible to everybody.