P&T Contact Form plugin: How to display error messages in other languages via Ajax
Well, the problem was that I was posting my Ajax request to / from all languages, therefore the site replied back with the site main language, which was French in my case.

The following finally works:

// Get the proper url to post to

if ( '{{ craft.i18n.getCurrentLocale() }}' == '{{ craft.i18n.getPrimarySiteLocale() }}' {
     var url = '/';
} else {
     var url = '/{{ craft.i18n.getCurrentLocale() }}/';
}

// Send it to the server

$.post(url, data, function(response) {
     .......
});