Why should each person have their own account?

Because pon identifies people by their account. If two people sign in with the same account (e.g. you on your iPhone and your partner using your login on theirs), pon treats you as one single person — and everything that depends on more than one person is lost:

• no „… is shopping now“ notification when someone heads out
• no telling who added or checked off what
• no live display during shared shopping

Here's how it's meant to work: each person creates their own account (ideally with „Sign in with Apple“), installs pon on their device — and then you share the list between your accounts (in an open list via „⋯“ at the top right → „Invite people“). Only then does shared shopping work properly, with all the hints and the live status.

Have you been sharing one account between two people? Create separate accounts and share the list anew — it's worth it.