Privacy
In short: everything you type stays on your Mac. The details below are checkable, the code is open.
The app
No telemetry. No analytics, no counters, no accounts, no crash reports sent anywhere.
Keystrokes. To know the last word, the app keeps what you typed since the last space in memory. It is never written to disk and never sent anywhere. The diagnostic log, if you turn it on, contains no typed words.
Network. The app goes online for exactly one thing: checking for updates. Once a day it reads a small file with the latest version number from GitHub, and when you agree to update, it downloads the DMG from there too. You can turn the check off in Settings. The Share buttons just open your browser, and only when you click them.
Clipboard. Corrected text is retyped, so the clipboard is not needed. In a few non-standard places such as Spotlight there is no other way: there the app saves the clipboard, uses it, and puts everything back right away.
macOS permissions. Accessibility lets the app fix text in other apps. Input Monitoring lets it see keystrokes. It asks for nothing else.
This website
The site counts visits with Yandex Metrica. It stores cookies and sends Yandex information about the visit: which pages were opened, where you came from, your approximate city, device type and browser. Session replay (Webvisor) is on: it records actions on the page, meaning scrolling, mouse movement and clicks. It does not record what you type into the feedback form; those fields are excluded. There are no other third-party scripts on the site.
You can opt out with the "Don't count my visits" link at the bottom of any page: after that the counter no longer loads in your browser.
The site's server keeps a standard request log: page address, time, IP address and browser. It is needed to run and protect the site and stays on the server.
None of this applies to the RuSwitcher app itself: it still sends nothing anywhere.
The feedback form sends what you wrote and your IP address, which is used to filter spam. Only the developer receives the message. It is never published or shared.
Open source
You can check all of this: RuSwitcher's source code is on GitHub under the MIT license.