Push messages (function menu -> Push messages) serve the purposes described in the article: https://qa.h-mdm.com/17024/
They should not be displayed on devices.
The Messages plugin (function menu -> Messages) is used to deliver a message to a device, a group of devices, a configuration, or all devices. Requires the Pager Plugin (com.hmdm.pager) application. Please make sure that the configuration you are testing contains this application. It is this application that displays messages on the device.