If you want piece of mind, buy off a dealer (well, most of them) and pay the price accordingly.

If you want more value for money up front - with the associated higher risk if it goes pear shaped; go private and good luck :)

I would never buy (and have never bought) a "modern" car worth more than ten grand privately, purely because you have too much to lose if things do go bang... but each to their own, everybodies threshold, and situation, is different.