Your text stays on your device. In Chrome, translation uses Chrome's own built-in, on-device translator (Chrome may download its language packs from Google the first time; your text is not sent). In other browsers, translate.gripe downloads an open translation model from this site once per language pair, keeps it in your browser's cache, and runs it on your device. Either way, this page sends no text anywhere.
Honest note: the open models are decent but not Google-quality, especially for less common languages. English to Japanese, Polish and Turkish use a many-language model and are rough. Pairs without English (French to German, say) go through English, which loses a little more. Check anything important with a native speaker.