Not necessarily. Bitgo uses a 2-of-3 scheme, where they provide retail account holders with a primary key, and a backup key. In the usual flow, you and Bitgo sign a transaction. If you need to recover, you can use your backup key.
Source: I am a Bitgo retail customer, but I don't have access to an enterprise account and cannot comment on their Bitstamp integration.
Source: I am a Bitgo retail customer, but I don't have access to an enterprise account and cannot comment on their Bitstamp integration.