Ie. they would be a 'dumb' storage system for the customers encrypted data. The data would be encrypted by the customer before upload. And then decrypted again by the customer after download.
Users would be identified by a unique random ID, and users would auth by signing a challenge with a secret key known only to the customer.
That way, even if a bad guy worked for lastpass and had full admin access to all servers, they couldn't steal anything.
And, in fact, if this system was properly designed, it could all run with opensource server code, and the datastore fully open for anyone to inspect, to prove that the security is down to cryptography rather than trusted yet fallible humans.