Answer
This works in private AI chats and People chats. Each tap updates the checkbox immediately, even if an earlier change is still saving. Changes to the same item save in order; a failed save restores the last confirmed state.
A confirmed check by someone else can notify the original requester, subject to chat mute, membership and device settings. Your own checks and all unchecks are silent. Unchecking and checking again creates a new completion event; retrying the same change does not duplicate it. Chat-list activity can update independently of push delivery.
Checking does not change the item order or whether the message is expanded. Checks, views and completion alerts do not call AI or add usage. Forwarding makes an independent checklist with its checks cleared.
On updated builds, if two participants check the same item, the saved check stays in place without a false error or another completion alert. Opposing changes can still require a retry.