Not all notifications require the network. Can't be more specific without knowing what notifications you are talking about, but stuff like calendar notifications, system messages etc only depend on information on the phone so turning the network off will make no difference to them.
Turn off notifications within the app itself, or use the system notification settings to block notifications from particular apps (assuming you are using at least Android 5). You can also define a quiet period in which only priority interruptions are allowed (Android 5 onwards), see the sound and notification settings for this.