I find it varies very heavily with the site and how I interact with it.
For sites I care about but don't actively keep up with, I want emails - they're slightly more sticky than push notifications.
For sites I care actively about (e.g. at least weekly visits), I want notifications on the site, not push notifications.
I am aware I am not most people in that I'm reasonably happy to proactively check sites I care about for updates. My experience is that many people like push notifications particularly if they're consuming content on their phone but that it's slightly less important if they're going to spend any serious effort responding (e.g. if you're going to put in 500+ word responses, push notifications is generally less interesting)
Note that push notifications requires either the hurdle of an actual app in a lot of cases, or getting people to pin the site to their device in the form of a PWA which has shown to confuse users/be a reasonably high friction exercise because it's not how people are currently used to interacting with a site and especially on Apple has some issues if you're a Chrome user.