Schedules can currently only be shared across different machines through taking advantage of the "Share" feature of Hydrant. My current setup is editing a Google Keep note with the URL whenever I edit my schedule, which is hilariously scuffed and error-prone.
This feature, if implemented, would allow a user to save their schedules to a database by logging in with their Kerberos identity, making them accessible across multiple devices. Saving schedules to local storage should still be an option for those who aren't MIT students.