OK, I think I found a partial answer to my own question after Mike helped to word the right question :) The eQSL.cc FAQ says:

How do I handle SWL cards? When a user sets the SWL/HAM checkbox in his Profile to SWL, it automatically forces all incoming and outgoing cards to say 'Confirming SWL Reception' instead of the usual 'This Confirms Our 2-Way QSO'. If you are an SWL, just make sure to go to My Profile and set yourself up as an SWL. If you are a ham, when you send an eQSL to an SWL, it will automatically put the correct wording into the card.

Though I still wonder whether SWLs send paper QSL cards, and if they do how such QSL cards supposed to be handled.

OK, I think I found an answer to my own question after Mike helped to word the right question :) The eQSL.cc FAQ says:

How do I handle SWL cards? When a user sets the SWL/HAM checkbox in his Profile to SWL, it automatically forces all incoming and outgoing cards to say 'Confirming SWL Reception' instead of the usual 'This Confirms Our 2-Way QSO'. If you are an SWL, just make sure to go to My Profile and set yourself up as an SWL. If you are a ham, when you send an eQSL to an SWL, it will automatically put the correct wording into the card.

Also according to this discussion SWLs do send paper QSL cards.