When more than one person is in your household, the list shows who did what. Something another person put on the list says Added by and their name. Something they’ve picked up says In name’s cart. Your own items don’t say anything.
While the list is open and someone else is changing it, their changes show up within seconds, so you can split up in the store.
If you both have things in the cart, tapping Clear asks Clear My Items (just what you picked up) or Clear Everyone’s Items. Anything left stays in the cart for the other person to clear.
The name is the profile each person picked when they joined. If they haven’t renamed their profile, the name comes from their email address.