Signing in attaches your Bluesky account to the profile you are using (or opens the profile it already belongs to): its watchlist stays the same on all your devices. What is synced (the records you watch here and your Discogs username) is saved in your Bluesky account, where it is public. Without signing in, everything stays in this browser.
A published list is saved in your Bluesky account, where anyone can see it, here on your public page and elsewhere. Turning it off deletes it from your account.
Your public page
Each profile has its own watchlist, Discogs lists, radio and filters, in this browser. A profile is local unless you sign in to a Bluesky account with it. Switching profile reloads the page. Anyone using this browser can open them.
Export the profile you are using as a file, then import it in another browser: no account needed. Importing adds its records to the profile in use.
Prices and stock come from each shop and can change. Links go to the shops.