Is it necessary for the mahram whose presence is stipulated during the meeting of the engaged couple to be from the fiancée's side? And has any Prophetic حديث (hadith) or story about the Messenger of Allah, peace and blessings be upon him, or his Companions been narrated regarding this?

The intent of a mahram (definition) is to negate the private seclusion (khalwa (definition)) between a suitor and his fiancée. This can be achieved in one of two ways: either by the presence of a mature, adult, male mahram (someone to whom marriage is perpetually forbidden due to kinship, suckling, or marriage affinity; a pubescent youth is considered as such, but not a child), or by the presence of a trustworthy and upright woman. The evidence for this is the Prophet Muhammad’s (peace be upon him) hadith (definition): "No man should be in private seclusion with a woman unless there is a mahram with her."