Can a process read its own "standard out" stream?

Yes, You may use os.Pipe() then process it yourself:

tmp := os.Stdout
r, w, err := os.Pipe()
if err != nil {
    panic(err)
}
os.Stdout = w

Or divert os.Stdout to a another file or strings.Builder.
Here is the detailed answer:
In Go, how do I capture stdout of a function into a string?