Retaining a session isn’t specific to Twitch but a general part of web application development. You could do it client or server side but that’s a decision for you to make. If you look at the source code for the Twitch JavaScript library, you’ll see that session storage is done via two methods: sessionStorage and cookies for older browsers.
The authenticated field that you’re looking for is stored by just seeing if a token exists, so you could store a boolean saying they’re authenticated if you get an OAuth token back.