It's all set this way. I can't understand why it doesn't work :(

Is there any client support or something?

It is settable under Settings -> Notifications. Many are system set and cannot be modified, but those that can be enabled or disabled are as follows:

Someone sends me a message on italki By email

italki Community Notifications By email

Friend Requests By email

italki Newsletters and Updates By email

Got it. Thanks a lot :)

Yes. You can raise a support request under the "Support" option, which appears just below "Settings" in the browser when you hover the mouse over your avatar icon on the top right corner.

By the way, you have to click to fill the radio button against each settable option in order to enable it. Leaving it unfilled disables the notification, and this is the default setting.