I'm not able to answer the question for the Go side of things, as I don't really dabble there myself, but the JavaScript side is hopefully a little easier, and you can potentially draw analogues to Go from there.
In JavaScript (depending on implementation), you are strictly limited to one thread, meaning a single stack. As all threads share heap space, and most JS implementations heap-allocate just about everything, and only keep references on their stack, all threads can access the data at the same time, regardless of where they're accessed from.
Async functions actually have very little to do with this. Specifically, async functions are just regular functions which inform the engine to get back to this later. They also access the same variables.
This is where Garbage Collection comes in. GC allows the source code to define variables, and have it reference-counted. When you call an async function, the variables are declared, meaning their reference counts are 0. It's important to note, that the variables are declared at the invocation of the function, not at the invocation of the program, meaning each call of the function allocates its memory afresh - you keep a new heap-allocated value for each call. This is essentially a closure. All functions define closures as they create a stack frame and all variables used within the closure have some defined way to access their value.
The reference-counting mechanism ensures that you have access to this data as long as you need it, and that you can ensure that despite the same variable name, you're actually referring to different memory.
To summarise, async functions don't do any magic in terms of managing memory, they just hold their own. In principle, this is exactly the same as calling a synchronous function, only that it runs out-of-order.
I presume Go does something very similar. In terms of multithreading across async functions, depending on whether you use a heap-allocated value, the same principle applies unless you explicitly share the memory, which Go advises against.