Do I get notifications if someone buys something from my basket?

The answer is, it depends on the type of basket you're using.
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In a gift basket, only participants (not the owner) are notified when items are marked as purchased. This way, your family and friends can coordinate and avoid buying duplicate gifts without you knowing what’s been bought. As the creator, you’ll still get a surprise when you receive your gifts!
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In a collab basket, if another member marks an item as purchased, all collaborators, including the basket creator will get a notification.
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In either scenario - the other person needs to proactively mark the item as purchased in the basket app (the green tick on the item page).