It really depends. My kids are 10, 6, and almost 3. With my oldest, I will often go with his opinion. He really loves school, and knows that it's a pain to catch up when he misses is, so he will really only ask to stay home if he really feels he's too sick to make it through the day. Of course, if he's running a fever, I keep him home regardless. Usually, if there are no other symptoms, I will send him even if he has vomitted once -- he's a puker, and will vomit if he coughs, laughs, or cries too hard, eats something with a texture that triggers it, or sometimes for no apparent reason at all. It's gotten a lot less frequent as he has gotten older, but if there are no other symptoms, I ignore it and assume he's fine.
My 6 year old I need to be a bit more careful with -- she'll want to go to school regardless, so I really need to watch her and make sure she's really feeling well enough, the same with my 2yo.
Really, I can usually tell when they're sick.... when they're really not themselves, I tend to keep them home, particularly in the younger grades.
Also, when the "smell sick." My kids all smell slightly different when they're sick.... not bad, but just different.... like their chemistry is off. Has anyone else noticed this about their kids?