Submissions are saved as a contact in a system-created list called Email Newsletter inside Marketing Campaigns, scoped to whichever kitchen location the visitor was browsing (Marketing > Lists). What happens after a successful submission depends on your Newsletter Popup - Show Behaviour setting:

• Show every time page loads: the visitor's browser remembers they subscribed, so the popup stops auto-opening for them.
• Show one time (the default) or Show when triggered (Footer button click): the popup's auto-open (or trigger) behaviour is unrelated to whether they subscribed, so it isn't suppressed by a submission alone.

In every mode, once someone has subscribed from that browser, clicking the Sign Up for our Newsletter footer button again shows You are already subscribed! instead of reopening the form.