What I've found is that on first dates people are on their best behaviour. Second dates people come out of their shells a bit more and some of their more hidden weirdnesses may come out. You may choose to not see them again after this. I'd save exclusivity for after a third date, possibly having gone to bed with them and finding out if they screech like a gibbon.
I wouldn't sleep with two people at the same time, but I would consider still having first/second dates with more than one person as, quite frankly, so many just disappear due to issues with exes, deciding they don't want to get serious, or the fact they don't like me screeching like a gibbon.
People who want to get exclusive straight away, without even meeting you (in some cases) - that's usually a red flag indicating possessiveness and insecurity.
WFF - we always notice you :)
Also, what counts as 'disappearing'? I had a date on Friday with a woman who just wasn't my sort. At all. It was obvious. For the first time ever I actually thought about just saying 'no' and turning around and leaving when I got there - I didn't, cos I'm not an arse, but it was so obviously not right I did think about it for a second. So I didn't mail or call her afterwards with a 'thanks but no thanks' - does that make me a disappearer?