no, unlike cars, trailers are pretty easy to get new titles for. you need to make sure it has no serial numbers on it, manufacturers name, etc. though. or, if it does, make sure you can remove them. you might just get a bill of sale from the seller to prove that you paid for it and what the amount was, but even that's not really needed.
if it's a trailer that was manufactured by a company or has been previously titled, you're in a bit of a gray area. technically and legally, you're supposed to follow the proper avenues to get a replacement/duplicate title from the last person it was titled to. that can be a pain if not impossible unless the person is readily available. however, you can easily play "dumb" and pass it off as a trailer that's never been titled before. it's next to impossible to not be successful. even if by some miracle you screw it up, all they'll do is deny your application. so, if you have knowledge it's been titled in the past, i guess it becomes a moral judgement call. do the means justify the ends so to speak. personally, i have no problem with it. the same goals are still reached either way. the state still gets their money, you've done the "right" thing by making it legal, you're able to register/tag it, it's able to be identified in the event it's stolen, and you're able to pass it off to a new owner in the future with no problems.
so, the process is as follows: you'll go to the dmv and tell them you have a custom/homebuilt trailer and you need to get it titled. they will give you a number and some paperwork to fill out. you will have to stamp or use weld beads to put the number on the trailer. they'll tell you where to put it. then, you'll carry the trailer and your paperwork to a dmv facility that has an officer (license place, weigh station, whatever) or anywhere you can find a state trooper. they will inspect it to make sure it's legal/roadworthy, check to make sure there are no existing serial/id numbers, and make sure you have put the new number on correctly. if all is well, they'll sign off on your paperwork. it's submitted to the dmv, you pay your title/tag fees, and you'll get your new title in a week or two.