It depends.  

For Earth OnDemand Users

Let us know if this is something you are interested in by voting for this feature.  We are contemplating a low-cost paid version of Earth OnDemand that would have this feature.

For EarthAI Notebook Users

We are adding the ability for EarthAI Notebook users to store their data in a private catalog on Earth OnDemand.  This feature will be available soon.

For organizations

Astraea offers paid subscriptions for Earth OnDemand for organizations that wish to manage their own data in Earth OnDemand or access Earth OnDemand via API from other tools.