When a card is shared, that card is available to both you and your co-parent. Only the parent who created the card has the ability to change or delete it. Both co-parents can see the card in their account and the Info Library Record.
Sharing a card does not create a new tab for your co-parent, but it will show the name of the tab in which your card is located. Sharing a single card within a tab does not share the other cards within that tab.
If a shared card is updated to private, that card is no longer available to the co-parent within their Info Library. However, the information in the card at the time it was shared will be permanently available on your co-parent's Info Library Record.