I can't speak to buying a car from a private dealer (I wouldn't do it), but I can speak to the Toyota "certified" used car/warranty. In my mind, it is worth it. Buying a used car is a gamble and this gives peace of mind. There's too much that can go wrong with a car, let alone a used one. Worrying about my car is the last thing I would want to do, so I would go for this type of warranty. While I wouldn't buy one of those extended warranties, the last 2 cars I have had came with 4 year/50K mile warranties and I was very glad they did. I found Toyota service/warranty, etc. to be great when I had one and I am actually thinking of going back to one for my next car.
2015 Trek Silque SSL
Specialized Oura
2011 Guru Praemio
Specialized Oura
2017 Specialized Ariel Sport