For EU citizens.
In principle, there are no border controls between Schengen countries (some temporary exceptions may apply). An ID is the minimum type of identification that may be required in case of control.
The first advice is to make sure you keep these documents safe while traveling, their loss is always a difficult moment and can in some cases lead to very complicated situations.
Airlines verify your identity for two reasons: to ensure that you are the passenger they have sold the ticket to and to ensure that you can actually enter the country you travel to as they could either receive fines or have to repatriate you in case of refusal of entry. Consequently Airlines tend to err on the “cautious side” and sometimes request more than actually needed. Their conditions of carriage should help you know what to do.
In case of theft or loss of your identity document, you should promptly report it to the local police who will issue a notice.
This notice could be sufficient for traveling, however we recommend to ensure it is the case by asking the customer service and keep track of what they tell you (preferably in writing) as cases of denied boarding still exist in such situations (different appreciations between customer service and ground agents).
A notarised copy of your passport will help you to obtain replacement but will not allow you to travel.
In any case, it is always better to contact your local consulate immediately in order to obtain a temporary travel document. As these are official documents, and despite having a limited validity, they will allow border crossing.
Once back you should follow the regular procedures of your country to obtain the replacement documents.
You should note that in most countries within Schengen once a the declaration of loss of an identity document has been filed, the document loses validity (don’t try to re-use it if you find it, send it back to the authorities of your country).
Also remember that a driving licence is not considered as a travel document. Passports are, Id cards also but only within Schengen if you are a EU citizen.