One thing that I love and hate is that my iPad, iPhone and iMac all ring/ding with every message. This is great when I'm out with a device and I'm sure to get the message.
It stinks when I'm at my iMac having a back-and-forth conversation with someone and all three devices are going off constantly.
The service should sense I'm chatting from one device and turn the notifications off on the other two devices. Once there is a delay from responding...all three device should re-alert of the last message. sent to you, if the other party (or the new party) were the last to respond.