Hi,
You can accomplish this multiple ways.
1. Create a landing page inside Adobe Campaign which is used to display recipient information. Later you can have this landing page displayed on your public website inside an iFrame element. Learn more about designing web applications at Designing a web application
2. You can have external application fire SOAP calls to campaign API and access any recipient's details. More details here About web services
Hope this helps.
Regards,
Vipul