When you implement xMatters they give you a a set of install instructions and some code to install on the Service Desk servers. It does work using Notification Methods yes. When a ticket is assigned the notification method gets called and the details goes to xMatters which then handles the interaction with the team.
Generally you would do it on a team by team basis and send all notifcations to xMatters for that team rather than just those of a specific priority. Then in xMatters the users can choose exactly how they want to be notified for each priority of ticket. They can also vary the notification method depending on time of day and you can set automated escalations etc.
SDM needs to be able to talk to XMatters (they need to be able to see each other using the Webservices APIs. So for this to work for mobile devices the mobile devices talk to xMatters and xmatters proxies the requests to SDM using the API and returns the responses to the mobile device. So the mobile users only need to be able to see xMatters not SDM. Its only xMatters that needs to be able to see SDM.
I hope that helps, happy to clarify if i've not been clear.
Carey