What solution do you use to send product notifications?

In our case, we just use email. For emails, in the past I used Mailgun because they have a very generous free tier, but ultimately I switched to AWS SES.

For marketing emails, we use hubspot or mailchimp. For transactional emails you can use SES AWS. I haven't used an SMS provider yet.

We are a big fan of loops.so for sending transaction and newsletter emails, emails look great, has a great editor and lots of configuration logic. Before that we used SES with semplates.io for transaction emails which was very cost affordable but not quite as flexible.

We use as well AWS SES for sending emails. SMS or Push is not used by our app. But here I would recommend AWS SNS or AWS Pinpoint