I do know that an EEA national needs only their passport to live and reside in an other EEA country, but do all EEA countries issue any piece of ID upon request of the interested EEA national?
For istance I'm talking about these cases.
France
Belgium
http://www.ixelles.be/site/250-Document-attestant-de-la-permanence-du-sejour-carte-E+
http://www.ixelles.be/site/686-Attestation-d-enregistrement-carte-E
At the moment I'm finding nothing about other countries, but wouldn't it be uncomfortable to be bound to use one piece of ID of another country in order to deal with authorities in a given country? Especially if the ID card is lost and the identifiee has to go to the consultate which might be far from him.