I've been playing with Pushover for sending Nagios alert push notifications to myself and other on-call staff. The Pushover service itself is free, until you start sending thousands of push notifications a month (hopefully for your sanity, that's not the case). Pushover can be configured with its own set of alerts, which you can make as annoying as you'd like. I'm sure there are many other push notification providers out there - I've just only had experience with Pushover, so that's the only one I can speak to with any authority.
The big issue with either SMS or email for notifications is that neither of them guarantee delivery within a reasonable amount of time. Email can (and does) frequently get hung up in various SMTP servers, and I've seen SMS delays on the order of weeks more frequently than I'm comfortable with.
With push notifications, there is a much stronger guarantee that the message will get pushed out to the user's device within seconds. In my testing with Pushover, the notifications always arrive in under 5 seconds after I hit their (very simple RESTful) API. There is always a possibility that your push notification provider's API is down, but you could always catch that condition in your notification script and fail back to email, SMS, or even something like PagerDuty (which provide a great service, IMHO).