Most private sellers aren't as experienced in negotiating as dealers, nor do they want to negotiate as car dealerships do. Use this to your advantage and make a fair but aggressive offer. If the seller turns it down, be persistent and counter with a slightly higher amount. Remember, it might be OK to spend a little more than you'd hoped if you found the perfect used car.
There's a big safety benefit to buying the newest car you can afford. When you buy a new car, you're ensuring you're getting a car that's up to the most recent safety standards in both crash testing and in safety technology. A new car compared to a used car from even five years ago can have big safety advantages with things like more airbags, a more rigid construction, and more advanced technology like a backup camera which is now mandatory on every new car sold in the U.S.