Ive seen people do both no bedding, and bedding. When we went to get my horse last year, we borrowed a trailer, and they just used loose hay as bedding....there might have been some shavings mixed in, but there was quite a bit of hay. And then one of my friends, whenever she traveled, she didnt have bedding. She has secure, grooved, non slip rubber mats. She went all over with protecting her horses too though.
All in all, like most things horse related, Id say its personal preference