Socialite is a, mostly, OAuth Client (as in it connects with an OAuth Provider (Facebook, Twitter, GitHub, etc) to validate that the person using your app has an account there).
From what you have described, I'm left with the impression that you are wanting to authenticate users that exist in for your application, in which case you will need an OAuth server. I have been using lucadegasperi/oauth2-server-laravel .
If you are actually using Socialite to authenticate users in this way it was intended you will need to store the token given to you by one of the OAuth Providers in your database. However, this code MUST NEVER leave your server with the exception of making a request to the OAuth Provider that gave it to you.