Problem with messages transation.

Where can we see the website in question?

It’s not a public for now.
How I can send you private link to this site?

Welcome to Support Forum — Please Read Before Posting

Okay.
This is a site, form page – https://wp.sicher-testen.ch/deutsch/kontakt/
This is a screenshot with attempt of submit an empty form – https://www.awesomescreenshot.com/image/36939962 with English error messages.

The locale set on the contact form is en_GB (English (UK)), so displaying English messages is correct behavior.

If you want to create contact forms in other languages, see Contact form in your language.

How can I change locale for the already existing forms?

In your link I see that I need to change locale for admin to language that I need, after that create new form and only after that we will have correct locale for new form in Frontend.
I know many strange solutions in many CMS, but this is one of strange!
If I have 20 languages for my site I need to repeat this actions 20 time?..
Why we can’t select the locale for currect form directly in this form – https://www.awesomescreenshot.com/image/36979934
In this select we will have a list of our languages – exactly same as for this “Users > Your Profile?admin menu page. You’ll find the language selector there.”