How do I get notifications from shared lists?

So you notice when someone else heads out or changes something, turn on the list's notifications: in an open list via „⋯“ at the top right → „List settings“ → „Notifications“ → the „Collaboration“ section. There you switch on „Share Shopping Start“ (then the „… is shopping now“ card and the push appear) and „Updates“ (alerts when the list changes).

Important: this requires that you've allowed pon to send notifications (iOS asks the first time). And it only works properly if each person has their own account — otherwise pon can't even know that „someone else“ is shopping.