LanguageTool
languagetool.org
Open-source grammar checking across more than twenty languages
grammarly.com
Catches the grammar and tone problems you stop seeing in your own draft.
The free tier fixes spelling, grammar and punctuation as you type, either in its own web editor or through a browser extension that works inside Gmail, Docs and most text boxes on the web. Its real value is the tone flag, which catches the sentence that reads as curt to someone who does not know you. The rewrites worth having are behind Premium.
Questions
Free to use, but you have to register before it will do the job. Free covers spelling, grammar and punctuation; clarity rewrites, tone adjustment and the plagiarism check are Premium.
Yes. Grammarly sends your input to its own servers to process it. That is fine for public material; for anything containing personal data, use one of the on-device tools instead.
Free covers spelling, grammar and punctuation; clarity rewrites, tone adjustment and the plagiarism check are Premium.
Alternatives
Same category or same task, ordered by how closely they overlap.
languagetool.org
Open-source grammar checking across more than twenty languages
quillbot.com
Rewrites a sentence you cannot get right into several alternatives
calmlywriter.com
A blank page with a focus mode that dims everything but this sentence
hemingwayapp.com
Highlights the sentences nobody is going to finish reading
onelook.com
Finds the word you mean when you can only describe it
wordcounter.net
Counts words and characters, and shows which words you overuse