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Allegedly Grammarly does not log your key strokes:
https://support.grammarly.com/hc/en-us/articles/360003816032-Is-Grammarly-a-keylogger-passwords.
Grammarly does not record every keystroke you make on your device. Grammarly accesses only the text you write using a Grammarly product, and only for the purposes of checking your text and providing corrections. Additionally, Grammarly does not process anything you type in text fields marked "sensitive," such as credit card forms or password fields.
Of course you can always find statements to the contrary:
https://techbeacon.com/security/grammarly-leaks-everything-youve-ever-typed-service-everything (2018 article):
Tavis Ormandy discovered that any webpage could easily hijack your session and steal all the information in your Grammarly account. And that includes absolutely everything you've typed into the service.
Although the above may have been addressed since then, it does go to state that nothing is free from vulnerability, so the less third-party whatever’s that can access your keystrokes the better.
We’ve also got the following to go by from reading Grammarly’s privacy policy:
https://www.grammarly.com/privacy-policy#how-secure-is-my-informationGrammarly is committed to protecting the security of your Information and takes reasonable precautions to protect it. However, Internet data transmissions, whether wired or wireless, cannot be guaranteed to be 100% secure, and as a result, we cannot ensure the security of Information you transmit to us, including Personal Data and User Content; accordingly, you acknowledge that you do so at your own risk.
See also:
https://www.quora.com/Is-it-safe-to-use-Grammarly