Maybe I'm paranoid too (and I am a Southerner!) but I have never let any of my children go on a play date without me at least knowing the parents a little bit first, ie by going round too on the first play-date.
Just because somebody has a child at school, doesn't mean they are necessarily 'safe' to look after my young child, IMO. If you wouldn't drop your child off at a total stranger's house, why do it with one just because they happen to also have a child? I know they are much less likely to be a wierdo, but you never really know till you see it all with your own eyes do you? Even something more 'minor', like if they smoke in the house, or have a huge dog, or an uncovered pond, is something you don't know till you've been there yourself.
My cousin was always a bit more laid back than me with all this, and when her son was ten there was a new boy at school, who invited him to a cinema party for his birthday. The plan was that all the mums dropped their children off at this kid's house, then the parents were taking them there & back in their car. Anyway, a couple of hours later my cousin had a call from the police, who had her son and two other boys at the local station. She was told that the father had taken the children in to the cinema, left them there, gone off and got drunk, came back at the end and tried to get the boys into the car to get them home. A couple of them wouldn't get in because he was so clearly p**d, at which point he got abusive and loud and tried to force them all into the car. At this point a member of the public called the police.
The man is now being prosecuted. I know you will probably all say that these things are rare, but my point is, things can go wrong, not everyone is a model parent, and I think taking the time to get to know somebody a bit before you let them take charge of your child is definitely worth it.
And yes I am a paranoid mother!