Basically, if you are using your iPhone the notification won't go to your Watch because, presumably, you've already seen it on your iPhone. If you are not using your iPhone, such as it's in your pocket or setting on a table, then notifications will only go to your Watch and your iPhone won't do anything.
It's not a bug, that's how it is supposed to work. Your Apple Watch alerting you by tapping your wrist replaces your iPhone alerting you by making a sound and vibrating.