In the Muwatta of Imam Malik, some ahadith start with "XXX related to me from Malik" - Who is this "me" that's narrating? Others start with "XXX related to me from YYY" with no mention of Malik.
What exactly is going on here with the chain?
The one who related from Malik in al-Muwatta is Yahya ibn Yahya al-Laithi يحيى بن يحيى الليثي when the Muwatta' is quoted without referring to a narrator. Yahya was the last known student of imam Malik who presented his version of al-Muwatta' to his teacher and got it approved. Therefore it is regarded as the final version of the book and the reference to all earlier ones. Yahya dictated his Muwatta' to his son and he heard al-Muwatta' first from Shabtoon شبطون before going to study in Medina and hear it from Malik himself or his students afterwards. As Malik used to leave a student reciting the hadith while he corrected them. Today we still have about ten versions of al-Muwatta' at hands out of more than hundred as some scholars say. Muhamamd Fu'ad 'Abdulbaqi محمد فؤاد عبد الباقي compiled and rearranged a Muwatta' edition that includes all these (accessible) Muwatta' versions.
So "me" is referring to the son of Yahya ibn Yahya if he heard the hadith from his father saying "Yahya related to me from Malik" or in cases "And he related to me from Malik وَحَدَّثَنِي عَنْ مَالِكٍ". While other statements must be analyzed separately based on the source you found it in. As for example in al-Muwatta' of Muhammad ibn al-Hassan a-Shaybani محمد بن الحسن الشيباني you may find ahadith which are related from his kufi and other teachers too.