As a long time arena participant, I prefer to have character sheets. Not only do I like character sheets, as reading material, but people aren't as trustworthy without them in my experience.

Without one, they can easily just start adding things in at whim because there is really nothing to regulate it, causing further scrutiny of character sheets to make sure everything fits within the world, and is acceptable. This can range from weapons and abilities to money. 

A character sheet lets you know what they have, and all of that so nothing is slipped in.