“I’ll Shout You a Coffee” Means “I’ll Pay”

In informal Australian English, shout can mean more than raising your voice. It can also mean buying someone a drink or some food. I’ll shout you a coffee is a relaxed way to say that the coffee is on you, especially between friends or colleagues.

This differs from I’ll shout at you. Shout someone something means paying for it, while shout at someone means speaking loudly or angrily towards them. You may also hear It’s my shout, meaning it is my turn to pay. For example: You got lunch last time. I’ll shout you a coffee today.

https://dictionary.cambridge.org/dictionary/english/shout