Why would you offload payments instead of e.g. connecting directly to payment gateways yourself, but wouldn't do the same for auth & notifications?
https://medium.com/airbnb-engineering/airbnbs-promotions-and...
https://engineering.linkedin.com/blog/2018/03/air-traffic-co...
What kind of notifications are we talking about where you can get away with self-hosting? AFAIK, Push Notifications on smartphones have to go through Google and Apple's servers. Maybe if you're doing email notifications?
For email though, not sure, that's a completely different topic. It comes with many headaches, using a service like SendGrid or Mailchimp seems like a good idea, and if you really want to roll our own mail servers down the line you can still do that.
These services are supposed to make things easier and have additional features like SMS notifications and so on, but, well not sure, never really used them myself.
For email we run bsd (not that that matters, it is just a factoid) servers, correctly configured which seem to work well (mail does not get to spam unless it is).
GDPR compliance is not something you can just buy.