Is this private?

Yes. To deliver notifications, the following things are stored: your HN username, notification preferences, a random device ID, and the push address and keys your browser provides.

How long does it take to receive a notification?

For immediate notifications, usually about 20 seconds.

Doesn't HN purposefully not have this feature to avoid flamewars?

If you feel predisposed to engaging in flamewars, you should consider not turning on notifications.