Right now they are independent. Your system prompts, settings, and most of all threads/folders are not shared.
Try something like how Brave browser creates private “sync chains”.
There's absolutely a way to do this. Even through encrypted cloud storage, which retains the same level of privacy, but greatly improved functionality.
It can be optional.
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Completed
Feature Requests
Backup / Sync
9 months ago

An Anonymous User
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Completed
Feature Requests
Backup / Sync
9 months ago

An Anonymous User
Get notified by email when there are changes.