You can control a number of behaviors for the iOS client and the way Notifications are sent and viewed including the ability to not play a sound (vibrate) or to prevent message text from being displayed.
Navigate to Settings > Amazon Chime > Notifications
Allow Notifications, when turned off, prevents any notifications from being displayed on the device.
Sounds, when turned off, results in the device vibrating for incoming notifications.
Badge App Icon, when turned off, will prevent unread count and other indicators from being displayed on the app icon.
Alerts can be blocked individually using the settings from the Lock Screen and in History.
Show as Banners, when turned off, prevents message previews from being displayed. When turned on, the user can choose between a Temporary display or Persistent which requires a swipe to dismiss.
The user can also choose what text is displayed in the alerts. Always (which is the default) will display message text, When Unlocked will prevent message text from being displayed on the Lock screen or Never which will display an alert indicating that a message was received but will not include any text.