Hi, Marius. Mosh for Chrome does store a few items, such as ssh fingerprints, in the
chrome.storage.sync API. This API stores data with Google for users signed into Chrome. This data is only visible to the app and is not directly visible to the developers. If you are not signed into the Chrome browser, it should be a moot point. If you are signed in, I would assume your company has approved such usages.
Mosh for Chrome has no provisions for sharing any data whatsoever with its developers. Note that the Chrome browser itself collects and shares usage statistics for all apps, which is available in an anonymized form to app developers through the Chrome Web Store Developer Dashboard.