I have been using LanguageTool[1] for years as "an open source alternative to [old school] Grammarly". It doesn't do that fancy "make this text more professional" AI stuff like this or Grammarly can now do, but they offer a self-hosted version so you don't need to send everything you write to OpenAI. If all you want is a better spelling/grammar checker, I highly recommend it.
You can also run your own local instance for the in-browser checking, which is handy for me as I need to be careful about sending text off to another company in another country (due to both client security requirements and personal paranoia!).
You don't get the AI based extras like paraphrasing, and the other bits listed in as premium only (https://languagetool.org/premium_new), but if you install the n-gram DB for your language (https://languagetool.org/download/ngram-data/) I found it at least as good as, for some examples better than, Grammarly's free offering last time I did a comparison.
It's great. I had a subscription for Grammarly for a couple of years and used both tools in parallel, but found myself mostly using languagetool increasingly. It is strictly better, I'd say even for English but certainly if you need other languages or deal with multilingual documents. So I canceled Grammarly and didn't miss it since.
You also can self-host and we do that at my workplace, because we deal with sensitive documents.
For VSCode users who want to try out LanguageTool, I cannot recommend the LTeX extension [1] highly enough. Setting up a self-hosted configuration is really easy and it integrates very neatly with the editor. It was originally built for LaTeX but also supports Markdown now.
And you can write your own custom rules. It's great as a reward for spotting an error in your writing you get to write a tiny little bit of code to spot it automatically next time. I've collected hundreds.
Absolutely plus one on this. LanguageTool is great and I’m also very happy on the free tier. With the app installed on macOS it also checks mails in the Apple Mail app, for example.
This explains why I was confused by this. I moved to LT many, many years ago, and didn’t know about those new Grammarly features. So I really wasn’t clear how rewriting a specific text had anything to do with Grammarly.
And if you are in a regulatory environment (or elsewhere where data exfiltration paranoia is part of your daily work life), you can install your own instance of the service (sans premium features) and not send your text anywhere outside infrastructure you control.
[1] - https://github.com/languagetool-org/languagetool