I am not sure about the threading problem, but from experience I can give you some idea about SSE
SSE's
Most importantly, the SSE depends on Javascript for a long poll , that is, it will continue to send requests to your server every few seconds, in an attempt to "listen" for updates
Check this when setting up the SSE eventlistener - it will send requests every couple of seconds to your server. The connection will not last forever
There is a great discussion here: What are Long-Polling, Websockets, Server-Sent Events (SSE) and Comet?
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The problem here, at least, is that if you send persistent requests to your server, it will not only be inefficient, but also very compressed in terms of functionality
As in pub / sub pattern , if you want to deliver live updates to your users, first you want to have some authentication (which, in my opinion, SSE does not provide), as well as ensuring that you have certain โchannelsโ for these users for receive updates for them
Having said that, there is a great post about creating a chat application with SSE here :

I only ever used SSE to send updates for the whole site, simply because I would prefer to have one connection (websocket) that I can authenticate and populate with specific data
Pusher
We use websockets wherever we can
However, the trick is that you can use a third-party website provider, such as Pusher . I am in no way connected with Pusher; we used them for a number of projects - EPIC service
Using Pusher, you can receive and deliver specific messages to users on your site without having to set up your own websocket server. You simply connect to the Pusher service with the same JS (eventlistener) setting, which will connect only once to the Pusher service: 
You can then send updates to your users simply by clicking on the Pusher API. We pre-installed this using the analytics application, which is under construction here
To answer your question, I do not know how scalable SSE
I tend to lead โbigโ Internet companies (as a demonstration of how to do this โrightโ), and I have not yet seen any of them endorse SSE over websockets. Maybe I'm wrong, but they all prefer websockets