The information is transmitted via HTTPS to the carrier. Once your session on Leaky ends, the data is gone -- there is no data store.
EDIT: in fact, for a subscription based service, you could figure out the optimal pricing quite easily. If the annual price of your subscription were less than the typical insurance savings by having it monitored, you'd have a hell of a business model.
If so, you need to encrypt it first. Storing sensitive data in the user's session is akin to writing it to disk in plaintext.