Most users that are new to ZNC Push expect to receive a push notification any time they get mentioned in a channel or receive a private message. However, by default, ZNC Push is configured to only send notifications when it's likely that the user is either away from their computer, or not paying attention to the conversation in question. This will ideally prevent you from getting "spammed" with notifications every time you get mentioned or receive a PM, especially when you are actively participating in discussion.
If you don't like this default behavior, then take a look either at the conditions section of the readme, or read through the example configuration guide. ZNC Push is highly configurable.
In some cases though, you may not receive notifications because your preferred notification service is not configured correctly, or the service itself might be having issues on their end. The easiest way to verify that ZNC Push is configured correctly is by sending a test message:
/msg *push send Test message
If you see an error message like "secret not set", or something similar, then you should verify that you've entered the appropriate value for that option. The push services section of the readme has details on which options are required for which notification services, and what values are expected.