The Pushover App (https://pushover.net) is an example of alert sounds that are unique to them available in its settings on iOS so it is doable. You can even set a default sound and a High Priority Sound. It also has a pretty good API for building custom alerts and push notifications as well as a DND option and much more! Oh and there is also a desktop option and a plugin to use growl on a mac.
I use it for the notifications (via IFTTT) to get the ones only I want to have rather than them all go through the SmartThings App as my wife (yes still working on the WAF) and other family members don’t want to get as many notifications as I do from the system. The latency issue in IFTTT doesn’t seem to be an issue when using a recipe for this as I think ST pushes the information pretty quickly to IFTTT without it being a poll which usually is around 15 minutes or so thus pretty useless for instant alerts such as changing HUE bulb colors on an event. But for cases where your using the ST channel to run recipes its great.
Something else it gives the option for is Group Delivery (http://updates.pushover.net/post/58172624706/introducing-delivery-groups) which I am considering exploring for specific notifications of multiple people.
I have also used Boxcar for this but its not as reliable (but also has custom alert sounds within the app).