Arc-style tab switching for Chrome and Firefox

No. It has no servers, accounts or analytics. Your settings, the order of your tabs and their thumbnails stay in your browser's own storage. The privacy policy has the details.

The switcher is drawn inside the page you're on, so its script runs on every page to listen for the shortcuts. It doesn't read the page. In Firefox the same access lets it take thumbnails of background tabs.

Another extension probably has the shortcut already. tabstack's settings then show it as “Not set”, and its toolbar icon turns red if it's ⌘J. In Chrome, set it at chrome://extensions/shortcuts. In Firefox, record a new one in tabstack's settings. On Windows and Linux, Firefox also uses Ctrl+Shift+K for its Web Console, so if search doesn't open there, pick another.

Click tabstack's toolbar icon. The switch and search shortcuts are at the top. In Firefox, press Change and type the new one. Chrome doesn't let an extension change its own shortcuts, so there Change opens chrome://extensions/shortcuts, where you set them. Below are four switches: switch straight to the tab when search finds only one, include recently closed tabs in search, cycle through only the current window's tabs (search still covers every window), and show timings under the switcher. Changes apply at once, and follow you to other computers if your browser syncs its settings.

Browsers only let extensions use letters, numbers and a few other keys for their shortcuts, and / isn't one of them. So ⌘⇧K is the browser shortcut and works everywhere, while ⌘/ is one tabstack listens for inside the page itself. That's why it only works while a web page has focus, not on the new tab page, in the address bar, or on the browser's own pages. Change it under “Also search with” in the settings, to any combination with ⌘, Ctrl or Alt. Reset brings back ⌘/.

Browsers don't let extensions run on the new tab page, settings or the extension stores. On those pages ⌘J switches straight away, and the next tab carries on the cycle. ⌘⇧K opens search in tabstack's toolbar popup instead of over the page.

Yes, once you allow it: “Allow in Incognito” in Chrome's extension details, or “Run in Private Windows” in Firefox. Thumbnails of private tabs are never saved.

Yes. Ctrl takes the place of ⌘ in every shortcut.