All Together

At this point, we only need to start the broker to get a fully-functioning (in the happy case!) chat:


#![allow(unused)]
fn main() {
extern crate async_std;
extern crate futures;
use async_std::{
    io::BufReader,
    net::{TcpListener, TcpStream, ToSocketAddrs},
    prelude::*,
    task,
};
use futures::channel::mpsc;
use futures::sink::SinkExt;
use std::{
    collections::hash_map::{HashMap, Entry},
    sync::Arc,
};

type Result<T> = std::result::Result<T, Box<dyn std::error::Error + Send + Sync>>;
type Sender<T> = mpsc::UnboundedSender<T>;
type Receiver<T> = mpsc::UnboundedReceiver<T>;

// main
fn run() -> Result<()> {
    task::block_on(accept_loop("127.0.0.1:8080"))
}

fn spawn_and_log_error<F>(fut: F) -> task::JoinHandle<()>
where
    F: Future<Output = Result<()>> + Send + 'static,
{
    task::spawn(async move {
        if let Err(e) = fut.await {
            eprintln!("{}", e)
        }
    })
}

async fn accept_loop(addr: impl ToSocketAddrs) -> Result<()> {
    let listener = TcpListener::bind(addr).await?;

    let (broker_sender, broker_receiver) = mpsc::unbounded(); // 1
    let _broker_handle = task::spawn(broker_loop(broker_receiver));
    let mut incoming = listener.incoming();
    while let Some(stream) = incoming.next().await {
        let stream = stream?;
        println!("Accepting from: {}", stream.peer_addr()?);
        spawn_and_log_error(connection_loop(broker_sender.clone(), stream));
    }
    Ok(())
}

async fn connection_loop(mut broker: Sender<Event>, stream: TcpStream) -> Result<()> {
    let stream = Arc::new(stream); // 2
    let reader = BufReader::new(&*stream);
    let mut lines = reader.lines();

    let name = match lines.next().await {
        None => Err("peer disconnected immediately")?,
        Some(line) => line?,
    };
    broker.send(Event::NewPeer { name: name.clone(), stream: Arc::clone(&stream) }).await // 3
        .unwrap();

    while let Some(line) = lines.next().await {
        let line = line?;
        let (dest, msg) = match line.find(':') {
            None => continue,
            Some(idx) => (&line[..idx], line[idx + 1 ..].trim()),
        };
        let dest: Vec<String> = dest.split(',').map(|name| name.trim().to_string()).collect();
        let msg: String = msg.to_string();

        broker.send(Event::Message { // 4
            from: name.clone(),
            to: dest,
            msg,
        }).await.unwrap();
    }
    Ok(())
}

async fn connection_writer_loop(
    mut messages: Receiver<String>,
    stream: Arc<TcpStream>,
) -> Result<()> {
    let mut stream = &*stream;
    while let Some(msg) = messages.next().await {
        stream.write_all(msg.as_bytes()).await?;
    }
    Ok(())
}

#[derive(Debug)]
enum Event {
    NewPeer {
        name: String,
        stream: Arc<TcpStream>,
    },
    Message {
        from: String,
        to: Vec<String>,
        msg: String,
    },
}

async fn broker_loop(mut events: Receiver<Event>) -> Result<()> {
    let mut peers: HashMap<String, Sender<String>> = HashMap::new();

    while let Some(event) = events.next().await {
        match event {
            Event::Message { from, to, msg } => {
                for addr in to {
                    if let Some(peer) = peers.get_mut(&addr) {
                        let msg = format!("from {}: {}\n", from, msg);
                        peer.send(msg).await?
                    }
                }
            }
            Event::NewPeer { name, stream} => {
                match peers.entry(name) {
                    Entry::Occupied(..) => (),
                    Entry::Vacant(entry) => {
                        let (client_sender, client_receiver) = mpsc::unbounded();
                        entry.insert(client_sender);
                        spawn_and_log_error(connection_writer_loop(client_receiver, stream));
                    }
                }
            }
        }
    }
    Ok(())
}
}

Inside the accept_loop, we create the broker's channel and task.
Inside connection_loop, we need to wrap TcpStream into an Arc, to be able to share it with the connection_writer_loop.
On login, we notify the broker. Note that we .unwrap on send: broker should outlive all the clients and if that's not the case the broker probably panicked, so we can escalate the panic as well.
Similarly, we forward parsed messages to the broker, assuming that it is alive.