Android smartphone owners who have Android smartphones fully supported with the Connected Phone app, which comes with Windows 10, will be alerted to new incoming messages via on-screen notifications. You can set the level of detail of these notifications yourself. Open the Connected phone app and click on Settings in the bottom left. Under the General tab on the right, you will see the appropriate settings.
Here in the Notification snippets section, you can record the level of detail of the screen messages that pop up. You can choose to hide the entire content, show only the sender, or show a preview of the message. In the same window, you can choose whether to display badges on the tray.