[FEATURE] Desktop: option to hide Code sessions from other devices (privacy)

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

The Code tab sidebar in Claude Desktop now shows sessions from every computer signed in to the same claude.ai account. I use the same account on my personal home computer and my work computer. My private sessions from home now appear in the sidebar on my work computer, where coworkers or IT could see their titles.

This is a privacy problem, not only a clutter problem. #92038 asked for a "this device only" option but was closed as not planned. I'm asking again from the privacy angle.

Deleting or archiving the session on the work computer is not a solution, because those actions sync back and also affect the machine where the session actually lives.

Archiving doesn't help either: archived sessions are still visible in the archive view and can be unarchived by anyone using the computer.

### Proposed Solution

Any of the following would work:
1. A per-device setting: "Show sessions from other devices in the sidebar" (can be on by default).
2. A per-session option: "Keep this session on this device only" (don't upload it to the account-level list).
3. A "This device only" filter in the sidebar, remembered per machine.

### Alternative Solutions

- Using separate accounts for home and work (costs a second subscription and loses shared settings).
- Using the CLI instead of Desktop for private work (loses the Desktop UI).
Neither is a good fit for a feature that changed behavior without an opt-out.

### Priority

High - Significant impact on productivity

### Feature Category

Configuration and settings

### Use Case Example

1. On my home computer, I create a Code session for a personal project.
2. At work, I open Claude Desktop on my work computer with the same account.
3. The personal session title appears in the work sidebar under "Other".
4. With the proposed setting, I would turn off cross-device sessions on the work computer, and personal sessions would no longer appear there.

### Additional Context

Related: #92038 (closed as not planned), #94262
Claude Desktop version: 1.52386.6（612f3f）
OS:Windows 11

Begin with the Claude Desktop Code-tab sidebar and cross-device session-list behavior described here; the issue names no repository files or tests. Done means an agreed per-device or per-session privacy control prevents private sessions from appearing on other devices without syncing destructive actions.

Written by the indexing model from the issue text.