Trailers are treated just like automobiles. The state of La issues a title. Here's what you need to do. You and your friend go together to a title transfer office (they are all over) You need a bill of sale from your friend to you, the title place can notarize this. Then, you pay $50 for a duplicate title and then the transfer fees and sales tax on the trailer. The title company will get a duplicate title in the name of the seller, then transfer title to you, you will get a new title in the mail approximately 2 weeks later. At the time of transfer, the title company will give you a registration and license plate on the spot in most instances.
It will help if you have the old license plate with the registration numbers on it. If not, there is a VIN tag on the trailer. The title company can pull up the registration info by VIN.
You may spend $20 or so more by going to a title transfer place, but you won't have to wait in a DMV line.