Short version: the text you translate goes to a translation engine, and nothing else leaves your device unless you turn on sync. The details are below; the full privacy policy is written to be readable too.
Text you translate
Translation happens on the servers of the translation providers behind each style. They receive the text and the language pair, nothing about you, and their terms don't allow them to use the text for training.
Pronunciation, transcription, synonyms and noun genders are fetched from Mate's own server, again with only the words involved.
Offline translation
With offline mode on, translation and text recognition happen on the device. Nothing is sent.
Web pages
The Safari and browser extensions only read the text you select, or the page you ask to translate. They don't collect browsing history. The "access to all websites" permission exists because Mate has to run on whichever page you're on. See Why does the extension need access to all websites?.
Camera
Photos you translate on iPhone are sent to a text-recognition service when online, and processed on the device when offline. They aren't stored.
Your account and sync
Without an account, History and Phrasebook stay on the device. If you create an account and switch on sync, they're stored on Mate's servers so your other devices can fetch them. You can view them at matetranslate.com/account and delete everything at any time. See Delete your account and data.
Passwords are stored hashed. We can't read them.
Payments
We never see card numbers. Apple handles App Store payments; Paddle handles purchases made from us. See How secure is my payment information?.
Analytics
The apps send anonymous usage events, such as which features are used and crash reports, so we can fix what breaks. They don't include translated text.