How can I tell at a glance that a list is locked, and why is the push button missing on its to-dos?

A locked list shows an amber closed-padlock pinned next to its name in the sidebar (unlocked lists only show a faint open padlock on hover). Inside the list, its name carries a small amber lock, and each to-do shows a small amber padlock where the push button would normally be — by design, since a locked list never pushes to your task apps.