Push notifications are best for notifying readers of new blog posts, content updates etc. Spamming people with offer after offer through push just pisses them off and they end up blocking all push notifications.
Use push to bring people back to your site. Not spam them with affiliate links.
I've used it in the past...You'll need lots and lots of subscribers to ever make much from push notifications, and it can cause problems with some antiviruses not liking your site (at least, with one network I tried it did)...I tested push notifications on my site for about 6 months and for me personally, it was better to just not use them..
We tried it a few times with nothing significant to gain. The fact is push notifications are more important for services like Facebook, Instagram etc. where one spends more time on their phone. The more you do something, you are more likely to get notifications. It is never going to be the case with blogs/sites. It is more important to get people to come back to the site more frequently and interact with the blog rather than getting a push notification. So we didn't find it useful for our site.