I haven't tested it myself, so I don't know for sure, but I wouldn't be surprised if push notifications don't work through the app, but regular alerts should work through the app, like they did before we had push notifications. Push notifications do work through Chrome on android, though. iOS...
I'm not sure if it works on the app, but I know it works on Chrome. My phone recently stopped working after a restart, so it's out of commission for now.
One of the features included in the new software version is the ability to recieve push notifications, which are notifications sent directly to your device, informing you of alerts and conversation messages. If your device is supported, you will get a prompt at the bottom of your screen asking...