Again, there are no rules here on which is better but price is the main factor; if the car is still fairly new and more expensive, a dealer will be able to handle that transaction much easier than a private seller.

Things to consider are; if you think that buying a cheap car from a dealer gives you more protection, in reality it doesn't. If the car is older with higher miles and maybe not a perfect history, there are very few warranties that will cover you if the car suffers a catastrophic failure. This doesn't mean the dealer is 'dodgy', it's just the risk anyone takes on a cheap car. Buying from a private seller has it's risks but you can at least gauge them, their house, their other cars maybe and decide if you feel the one you're looking at has been well cared for.

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Where the cars came from:

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This is a slight side-note but factors into the above. A main dealer will have their brand news cars and a selection of used vehicles, normally a couple of years old. These will have been taken in part exchange when someone came in to buy their next new car once their current one was 'old'. The money involved would make is easier to perform the whole transaction at the dealer.

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Those dealers will also take in cars, as part exchange, that are a little older than they'd like to retail. These will normally get farmed out to smaller, non-franchise dealers.

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Both of these 'larger' dealerships will also take in cars that sellers had frankly given up on; these are the cheapest, potentially worst cars. As any respectable dealership will not want to take on the risk of selling a car that's more likely to have problems, they farm these off to auction. In my view, any used car site that deals in sub ~£2000 cars is best to avoid.

That's not to say that cheap cars are to be avoided! But if you are shopping in that area of the market, buying privately and seeing the car in it's natural habitat so-to-speak gives you a much better chance of finding a good one.