Looking from the stern inside the external rubber flap, I can see an internal metal flap, freely moving on his asymmetrical axle, before the one activated by the switch. Obviously meant as an additional protection for water backflow, I guess.
But why only on stbd tube? No sign of it on port side...
Is there any weird reason that I'm missing, or is it just a wrong installation that I should fix to avoid troubles?

It is possible it blew out. I had one missing and had to replace it.

Aha. I didn't consider that because looking inside the port tube I couldn't see any sign of it, but I didn't have a torch handy.
How should it be fitted? Can be done from the outside?

You have to replace the tube. I bought mine from Rex Marine

Urgh! Sounds expensive, I'm tempted to remove also the other one...

For the records (and just in case there's anyone else interested), actually it isn't so expensive, and it's also possible to purchase the flap alone.
This is one of the exhaust flaps I found browsing around a bit, just as an example.