What's the best way to make sure the title for the out of state horse trailer I'm buying from a private party is clear?

Most states have DMV that issue the titles.  I would assume that you could check with them to see if the trailer has a clean title.  Is there are liens on the title. You could cut two checks, One to the first lien holder and the remainder to the owner for his equity.

This is usually reflected on the title itself. But there have been a few cases in the news of people haing bought cares with clean titles to ony find out that a duplicate title had been issued with a lien. So a call or visit to the DMV would clear that.