Starred notes are synced between devices.
If you are using Obsidian Sync you need to enable syncing of “Core plugin settings”, with other services it should be synced automatically.
After the file has been synced where this is stored you need to restart Obsidian to be able to see it there.
This is the same as with any other plugin data, it’s only loaded from file when Obsidian starts.
We might rework this behavior in the future to make it more sync friendly.