How would I go about streaming audio FROM an iOS device to another device like a Mac or another iOS device? I can only find solutions like AirPlay on Google and nothing abou
You say that you don't want to use AirPlay. Going with that, you will need to run your own server on the iOS device. This is certainly possible, and there are apps that run various servers for various tasks. However, I'm not very familiar with the subject, and it seems that you are correct that there isn't much information on this online.
That said, I'd suggest having a look at Darwin Streaming Server. It seems that it's a little old, but it's open source and might be of use to you.