I am usually aware the second I get a urge and my diaper fills, but then I immediately forget about it. Sometimes I can barely feel it happening and sometimes it's more noticeable. There are plenty of times when I'm not sure how wet I am or if I am even wet or not. When I am sleeping, I am not aware.
I don't think anyone notices I have wet because it just happens in any type of circumstance or activity and, unless my diaper is leaking, I don't pay any attention to it. A messy diaper is another story and when I have one I make my way to a bathroom to change. Most people close to me know about my IC issues, but don't say anything to me. My wife does not pay any attention to it either and, luckily, she has a poor sense of smell.
When I was younger and did not wear diapers as often, accidents in normal underwear did get noticed by others. Waaaay more embarrassing.