You can automatically share child cards with other members of a parent card – it is easy to spot a card that's shared within a parent as the parent badge will be blue.
Shared cards can be seen by everyone who is a member of that parent. If you're the only member of the parent, you are the only one who can see it!
Private cards can only be seen by those users you have specifically added to a card. If a user joins the parent they won't be able to see this card (i.e. will not be automatically added as a member to this card). Also members of a parent who are Readers are only able to add unlisted cards.
A Contributor, Editor, Moderator and Author of a card are able to both 'share' or 'privately' add a to a parent. Whereas Readers are only able to add 'private' cards.