It looks like there is an issue with Hangouts app that causes notifications for new conversations with a given contact to be disabled by default, with no way to override this setting. On the Conversations tab, these are denoted by a bell with a strike-through:

(click for larger image)

Once you open such a conversation you will see a message in red saying "Notifications off - You won't be notified of new messages" with an option to enable them:

(click for larger image)

Existing conversations for which you've enabled this setting will receive notifications normally. If you delete the conversation, the setting will revert to "off". Archiving the conversation keeps it enabled.

The details that are described in this thread (I suggest starring it to show Google that there is interest in this issue) match exactly what you are experiencing. It has been brought up before, and was escalated to the Hangouts development team. We can only guess when (or if) Google will fix this.

It looks like there is an issue with Hangouts app that causes notifications for new conversations with a given contact to be disabled by default, with no way to override this setting. On the Conversations tab, these are denoted by a bell with a strike-through:

(click for larger image)

Once you open such a conversation you will see a message in red saying "Notifications off - You won't be notified of new messages" with an option to enable them:

(click for larger image)

Existing conversations for which you've enabled this setting will receive notifications normally. If you delete the conversation, the setting will revert to "off". Archiving the conversation keeps it enabled.

The details that are described in this thread (I suggest starring it to show Google that there is interest in this issue) match exactly what you are experiencing. It has been brought up before, and was escalated to the Hangouts development team, who are already aware of the issue. We can only guess when (or if) Google will fix this.