I'm answering myself - in case someone has the same problem.
I had to disable the menu options that are presented with my theme (Suffusion) - after disabling the theme options and enabling only the polylang options, menus switch ok.
The only thing I've lost this way is the ability to show a different post title in the menu buttons (in my theme options, I could do it - e.g. in case of the whole post title 'upcoming events', I could show just one word 'events' in the menu bar. Maybe that's something the polylang team could include in the plugin in the fiuture.
Happy New Year!