By default, no notification or note is sent to the receiver when a gift is purchased for them. There is a “Do not spoil my surprises” setting among the features. You can opt out of receiving a notification when someone has purchased a gift for you. The feature is great for a surprise, but it can also result in minute hassles where the recipient ends up ordering the item from their wishlist while the gift is still en route.
Go to the “Your Lists” menu, then click on “Manage list” on a specific list that you desire to edit, then uncheck “Keep purchased products on your list” to prevent this. At the end of the day, this is your call to make. Select “Save Changes” at the end of the process.