One year I had a small 5x10 and a 38' RV. I pushed the 5x10 to the back corner of the site and back up close to it. No one from Disney gave me any grief, but it did take up some site space. The next year was a 8x15 to carry two carts, we parked that in overflow and Disney was more then happy to print us a second parking pass to tape to the trailer. I would say that using the overflow lot is the better option then keeping your trailer in the site if you use the site a lot. Heck last year I couldnt fit my cart and truck in my site, so I parked the truck down at overflow for the stay, considering doing that again this year as I just had so much more space.
We have in the past a few times parked our small trailer behind our RV on the site, premium site I do believe you are allowed. But be careful some of the loops are tight for getting your RV and trailer around in the loop. Loop 100 is a tight loop. Since now we always bring our large trailer we park in the parking lot, this is the better option.
As folks said you can keep it on your site if you wish but all things (cars, rv's, trailers) should be on the paved driveway or sand pad. None of those should be outside the pad timbers or off the drive. WDW doesn't want your campsite looking like Delta House.
If you do they might send a Cast Member like Scupper here to ask you to move it.
If it ends up in the overflow lot I would recommend a hitch lock.
Ive seen them kept onsite, however I have to say it was kinda annoying last year, our neighbor had an RV, enclosed trailer(for decorations that they set up all the way to our paved pad) and two trucks, obviously it wouldn't all fit on they're pad so they drove and parked off pad. I'm not a strict rule type person but it wouldv'e been nice if they did, it did start looking a bit sloppy with all that goin on, n theyr'e dog yappng...
The small trailers are easy to keep behind you and in your site. Anything bigger then a 5x10 I would rather have down in overflow anyway. And once your at a 20' trailer, well its down in overflow no matter what.