[BUG] Cowork: expose execution mode (Local/Remote) as a visible per-project setting, and document what leaves the machine

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

**Where this happened:** Cowork tab, desktop app on macOS (Apple Silicon), project-based workflow

### Summary

Cowork projects run in two different execution modes, the mode is visible only as an after-the-fact badge on the project card, and there is no setting anywhere in the Cowork UI that lets a user choose it.

### What I observe

- A project I created earlier displays a `Local` badge on its card in the Projects list.
- Projects created since come up remote. Nothing in the creation flow offers a choice, and nothing explains that the two behave differently.
- At least one session started inside the project that carries the `Local` badge still executed in a cloud container. So the badge does not reliably describe where the current session runs — which makes it worse than no indicator, because it reads as a guarantee.
- The only candidate control is Settings → Cowork → **"Only on this computer" ("Stops when the app closes or this computer sleeps")**. That description matches local-execution behavior, but the setting never uses the words local, remote, cloud, or execution, and it appears in no help-center article. I could not confirm what it actually governs without turning it on and inferring from behavior.

### Why this matters

I do advisory work on a client's files through a connected folder. "Where does this run, and what leaves my machine" is a question I have to be able to answer plainly — to myself and to the client. Right now the honest answer requires reverse-engineering the product's behavior, and the one indicator the UI does give me (the `Local` badge) is not dependable.

The Code tab already solves this exact problem: *"Environment: choose where Claude runs. Select Local for your machine, Remote for Anthropic-hosted cloud sessions."* Cowork has the same underlying distinction and exposes none of it.

### Prior art

#76143 reported the silent switch to remote-by-default in July and asked for execution mode as a visible per-project setting. It was closed as not planned and carries a `stale` label; the underlying request was never addressed. I'm filing rather than commenting there because the thread is closed, and because the transparency half of this — documenting what data crosses in each mode — was not part of that report.

## Requested changes

Three specific, small items:

1. **Surface execution mode as a per-project setting in Cowork**, matching the Code tab's Environment picker (Local / Remote). Show it at project creation and in project settings, not only as a badge after the fact.
2. **Document "Only on this computer."** If it is the local/remote control, name it as such in the UI and in the help center. If it is not, say what it does.
3. **State plainly what data crosses in each mode** — what stays on the device, what transits Anthropic's servers, and what a connected folder does or does not upload. One paragraph in the architecture overview would settle it.

Item 1 is the one that matters most; items 2 and 3 are cheap and would have saved me the diagnosis entirely.

## Environment

- Claude for Mac version: 1.46388.4 (50e62f)
- macOS: 26.6.2 (25G83)
- Apple Silicon
- Date of report: 2026-09-08

### What Should Happen?

Cowork should let me choose where a project runs, and should tell me the truth about where it is running.

Concretely:

1. **Execution mode is a visible per-project setting** — Local / Remote, shown at project creation and in project settings, matching the picker the Code tab already has ("Environment: choose where Claude runs").
2. **The badge on the project card reflects the actual execution location of sessions in that project.** If a project is marked `Local`, sessions in it run locally. If the badge can't guarantee that, it shouldn't be shown.
3. **"Only on this computer" states what it governs.** If it is the local/remote control, name it that way in the setting and document it. If it isn't, say what it does.
4. **The docs state plainly what crosses the network in each mode** — what stays on the device, what transits Anthropic's servers, and whether a connected folder's contents are uploaded.

Today none of these hold: the mode is unchosen, the badge is unreliable, the one candidate setting is unnamed and undocumented, and the data-flow question can only be answered by inference.

Item 1 is the one that matters most; items 3 and 4 are cheap and would have saved me the diagnosis entirely.

### Error Messages/Logs

```shell

```

### Steps to Reproduce

1. Open Claude for Mac → Cowork → Projects. Observe that some project cards carry a Local badge and others carry none, with nothing in the UI explaining the difference.
2. Create a new project. Observe that no step in the creation flow offers a choice of execution mode, and no project setting exposes one afterward.
3. Open a project that carries the Local badge, connect a folder, and start a session.
4. Ask the session where it is executing. In my case it reported running in a cloud container, not on this machine — despite the project's Local badge.
5. Open Settings → Cowork. Observe that the only related control is "Only on this computer," described as "Stops when the app closes or this computer sleeps." It never uses the words local, remote, cloud, or execution.
6. Search the help center for "Only on this computer." No article documents it.

### Claude Model

Opus

### Is this a regression?

I don't know

### Last Working Version

_No response_

### Claude Code Version

2.1.74 (Claude Code)

### Platform

Anthropic API

### Operating System

macOS

### Terminal/Shell

Terminal.app (macOS)

### Additional Information

_No response_

Start in the Cowork Projects flow and Settings → Cowork, then compare their execution-mode behavior with the Code tab's “Environment” picker. Review the help center for “Only on this computer” and the architecture overview. Done means Cowork exposes an accurate per-project Local/Remote choice and documents the setting and data flow for each mode.

Escrito por el modelo de indexación a partir del texto del issue.