Notifications on Web Pages
I'd say at least half of the websites I visit ask to show notifications. I've never said yes.
Whenever I Google a web development question I have, lots of articles come up from all different sources. I often visit sites I've never visited before (or don't remember visiting), and many of the sites I visit ask to show notifications. There are two techniques I've seen - one is to just use the Notification functionality of the browser, the other is to ask the user first, and then open the browser's notification request bubble.
Here's the deal - I have no history with the websites I visit, so I'm never inclined to say yes. I could see visiting a website multiple times and letting them show me notifications, but these websites always jump on the opportunity to show notifications. Currently, there's no way to disable those annoying notification popups and deny the notifications automatically, but I have heard talk of adding a feature like that.
Here are some thoughts I have about notifications on websites:
Wait for intrest
My advice to website developers is this: don't ask to show notifications until a user has interacted with your website in some way - maybe clicking on a link to another page, adding a comment, or signing up for an account. I know I'd personally be much more likely to allow notifications once I've already evaluated the value of the website, and it would seem a lot less like spam.
Get rid of popups & banners
I've seen some websites display a banner asking if I'd like to receive notifications. Though I can see why you'd do that (you have control over what the request says, instead of relying on the browser's native API), it gets annoying when it covers part of the page on my phone, and especially if it's hard to close.
You could add a link somewhere on the page that says "subscribe to updates" without covering half the screen or add a small button in the corner of the screen.
Provide a distinct advantage to the user, something that sets it apart from other notification requests. Tell the user how often they will receive notifications, and stick to what you said.