I think there are situations that warrant both, and lots of companies do both. For example, Google has an alerts button in many of their apps that shows in-app

I think there are situations that warrant both, and lots of companies do both. For example, Google has an alerts button in many of their apps that shows in-app messages about new features or issues that need your attention. They also send mass e-mails for more important notifications like policy changes. You can give your marketing staff the power they want without any in-house development by popping https://www.intercom.io/ https://www.intercom.io/ into your app. It handles both in-app messaging and real e-mails, with a system for segmenting users so you can send messages to only the groups that need to receive them, e.g. only trial users or only those that haven't logged in recently.