Can double-pole breakers trip each side separately?
There is no need for confusion, one just needs to read the specifications!

You have 1p breakers, one protected conductors.
You have 2p breakers, two protected conductors.
You have 1+n breakers, one protected conductor, one switched conductor, polarity sensitive.
And 4p or 3+N for three phase with or without neutral protection.

Look at the symbols.

One protected conductor.


Two protected conductors.
 

These symbols are printed on the unit.