You'll see the status of the cart from the Shopping Cart Manager page, which you can find if you hover over the cart icon in the top right of your screen.
After you send the checkout to the adopter, either by texting or opening the checkout in a new tab, you will be automatically directed to the Shopping Cart Manager to view that cart's status. The adopter's pending cart will have an action for "Resend Checkout," in case that is needed for any reason.
When the adopter has finished the transaction on their end, their cart in the Shopping Cart Manager will update to have an action for "Deliver documents". That's your indication that they've completed the transaction. Please note that the cart will not completely clear out of the Shopping Cart Manager until you complete the document delivery step.