I've noticed myself being able to do this, but modulo the thinking part. I can think about at most one thing at once, but I can think about what I want to type and start my fingers on their dance to get it out, while switching to a conversation that I'm in, replaying the last few seconds of what the other party said, formulating a response, and queuing that up for speech.
I strongly believe that the vast majority of people are also only able to basically do that - I've never met someone who is simultaneously form more than one "word stream" at once.
Yeah I agree, my brain works in that rapid switching kinda way. I feel you can set up the subconscious one to do a task while your conscious brain works on something else. Trying to output two conscious 'word streams' simultaneously as you said, feels impossible.