Desktop: session group names sync across devices via account, sessions don't — no opt-out (privacy concern for isolated work/personal machines)

# Summary

In the Claude Code desktop app, **session group names appear to be stored server-side (scoped to the claude.ai account) and therefore show up on every device logged into the same account**, while the sessions themselves are local files that never sync. There is currently no way to opt out of this group-name syncing.

This has two consequences:

1. **Privacy / information leak across machines.** I use one machine for company work and one for personal work (deliberately isolated). Group names I create on the work machine — which can themselves be sensitive (project codenames, client names) — appear in the sidebar of my personal machine, and vice versa.
2. **Confusing UX.** Because sessions are device-local, the synced groups show up empty (or partially filled) on the other device, which looks like a sync bug.

# What I observed

- Two machines, both running the Claude Code desktop app, logged into the same claude.ai account.
- Sidebar session groups (names) created on machine A appear on machine B.
- The sessions inside those groups do **not** appear on the other machine (expected — they are local).

Poking around locally on Windows:

- Session transcripts live in `~/.claude/projects//*.jsonl` and the desktop app's session index lives in `%APPDATA%\Claude\claude-code-sessions\\\local_*.json` — all device-local, all prefixed `local_`.
- The desktop app's cached frontend JS references an API endpoint `/code/session_groupings`, and sessions carry a `session_grouping_id` field — which is consistent with group definitions being an account-scoped server-side resource.

# Environment

- Claude Code desktop app on Windows 11 (both machines)
- Same claude.ai account on both

# Request

Either (or both) of:

1. **An option to keep session groupings device-local** (or at least an opt-out of cross-device syncing of group names), for users who intentionally isolate work and personal environments.
2. **Scope groupings per device or per environment** rather than per account, so groups only appear where their sessions actually exist — this would also fix the "empty groups on the other machine" confusion.

At minimum, it would help to document that group names are synced through the account while sessions are not, so users know group names are not device-private.

Reproduce the behavior with two Windows machines and inspect the cached frontend references to `/code/session_groupings`, along with the local session paths and `session_grouping_id` fields described in the report. Done means group names are either opt-out/device-scoped so they do not leak across isolated machines, or the account-sync behavior and its empty-group implications are clearly documented.

Scritto dal modello di indicizzazione a partire dal testo della issue.