We have national media (German: srf, French/Italian: rts, Romanche: rtr), people consume that, and a few medias that have multiple language versions like 20minutes.
We also have a few language specific medias (German: NZZ, Tagesanzeiger, Blick, ..., French: Le Temps, 24 heures, La Liberté, ...), but I think most people consume Swiss media, especially when Swiss politics and local afairs are absolutely not covered by French and German medias.
The funny thing is that people know more about what is happening in the neighbouring countries than in the other parts of Switzerland. The "national" media is very divided and only covers French-speaking regions in French, German-speaking in German, etc. as if they were local media.
Switzerland is like that. I remember asking (in my best German) the person manning a ticket counter at Zurich train station if they spoke French once, if a look could kill I'd be dead, lol.
It gives the strange feeling that although they decided to create a country together they don't want to interact with each others unless absolutely necessary.
That's not it. Everyone in Zurich studies French at school, if you speak slowly to them in French, it's pretty likely that they will understand a lot of what you say (if they are familiar with the context of course). Similarly, if you speak slowly in German to a French-speaking Swiss, it's pretty likely that they will understand a lot of what you say (again, if they are familiar with the context).
It's just harder to speak in another language than to understand it, so if you ask someone to speak...
Next time, ask them if they understand French :-).