This issue should be tested in all devices. In web, login in a site with a certain user (we'll call him u1). In the app, login in the same site with a different user (we'll call him u2). In the app as u2, go to messaging and open the conversation with user u1. In web as u1, open the conversation with user u2 and send him a message. In the app as 2, wait for the message to appear in the conversation. Check that you can see the message. Repeat steps 4&5 several times and check that you can always see the message received.