Not hearing notification sounds on Apple Watch
The Apple Watch does not play notification sounds for any applications, regardless of the Silent Mode setting. It will only ever play sounds for alarms. This is not something we are able to override in our application.
For notifications to appear on your Watch, the Watch needs to be secure on your wrist, unlocked, and your phone needs to have its screen off. When in this mode, if you have Notification Mirroring enabled for the Pushover app inside the Watch app, the notification will appear on your wrist with no sound, but will vibrate if you have that enabled. If you are seeing notifications on the Watch with a large icon, please see this article.
To continue hearing notification sounds on your iPhone even when your Apple Watch is on your wrist, you will need to disable Notification Mirroring for the Pushover application, which will stop showing any Pushover notifications on the Watch. You can do this by opening the Watch app on your iPhone, then select Notifications, then uncheck Pushover.