What are the privacy concerns for using Discord?

I think it's best to start with the classic mantra:
If you aren't paying for the service, you are not the customer, you're the product.

It's easy to think that Discord isn't reading your messages or listening to your calls, because the utilize End to End Encryption. And this is a good thing for them to be doing. It means that no one can intercept the conversation, as it passes over the web. However, there is one glaring loophole, the data is decrypted by the Discord app on your device. Does the Discord app then send any/all of that data up to their servers? Probably not, but they probably also have the app scan it for keywords and categorize it so that they can upload that metadata about you to their servers. Also, for public Discord channels, you can bet that they are reading, scanning, and categorizing everything on those channels. The Discord app is also collecting as much information as possible about the device you are using it on.
From their Privacy Policty:

Information about your device. We collect information about the device you are using to access the services. For example, this includes information like your IP address, operating system information, browser information, and information about your device settings, such as your microphone and/or camera.

The ultimate goal of this is to use this data to build a customer profile of you and sell that profile to advertising firms. As for how bad this is, that's up to your personal level of paranoia. For most people, this is probably a reasonable trade off, most of the time. If you are not the type of person who needs to protect their privacy carefully (e.g. a journalist in a hostile government) and the conversation you are having isn't all that important (e.g. talking about a video game), then it's probably fine. But, if you are having a conversation which might actually matter or you are worried about a repressive government, then maybe pick something with a better privacy track record (e.g. Signal).