Private sellers tend to ask lower prices, in addition to not collecting GST. But they may be harder to deal with - some private sellers are really good, others are frankly idiots.
A reputable dealer is probably trustworthy (e.g., if you're buying off a Toyota dealer's used car lot). Some of the small USED CAR dealers can be dishonest, though. Also, don't assume that because it's a dealer the car deal is good. My first car was bought used from a lot by a friend (who I then bought it from), and I found out while I owned it that it was a rebuilt car from Alberta... even though it came from an Infiniti dealer, my friend never knew it was once written off!).
On that note, do run a CARFAX check. They're not always perfect, but then can eliminate some bad vehicles. And as others have said, get a mechanic to look at it... in my case, it was pretty obvious once you popped the hood that the car had been in a serious accident and put back together.
As to your search: I'm not sure you'll find a good Fit in your price range (when I was looking ~18 months ago there were very few used Fits for sale, period). But if you do, it's imo the best car in its class (full discloure: I own one). The Yaris is a good choice, too. Those would be my top picks.
Lastly: a brand new Accent starts at $14k, but for a while they were $9999 with incentives... I don't know what Hyundai is doing these days, but check out their deals and you might be able to fit a brand new Accent into your budget if you're so inclined.
