Can one make writings or log during the Android: Netrunner play? Or every player should use only its own memory?
Taking notes is not permitted, no.
From the current Tournament Regulations (v3.2):
Taking Notes and Outside Material
Players cannot take notes or reference outside material or information during a tournament round. However, players may reference official rule documents at any time or ask a judge for clarification from official rule documents. Official rule documents include all rules documents and inserts available on the Android: Netrunner page of our website, those found in an Android: Netrunner product, or any portion thereof.