Basically, I feel like private chat is underutilized. I will sometimes use it to try talk to people and they never notice it blinking. Then I mail then and say "Hey, go to private chat". Then I wind up forgetting about it in the middle of a conversation. 

Why not give an OPTION (toggle-able in the Manage > Options) for private chats to make a noise (maybe the same one as when a timer runs out? maybe a different one?) so that people know when you're private messaging them. 

Not sure if this has been suggested before; I tried to find a similar topic, but failed to do so.

(Advance apologies if this already exists and I'm an idiot.)

raises her hand in agreement

I would love to see (hear?) this as well. Like Kathryn said, I've been having a little trouble with the same because it won't notify you unless you're actually in here lol. 

You're definitely no idiot, Kathryn . As far as I know this feature has not been added (yet) :P