I always worry that when you buy what you think is an ok car privately that it might break down the next week...but I guess thats the risk you take for a cheap car!
It is, but it's a risk you take buying a cheaper car from a garage too!
In my experience, if you buy from a garage or small-time dealer, they're more likely to know if something is wrong, but not more likely to tell you!
At least with a private seller, you can try to judge the situation and the person.
Try to get a service history, and if possible, get a mechanic to come with you..... the cost of an hour of a mechanic's time is well worth it for peace of mind!