There is no evidence on the legitimacy of a person reciting Qur'an on Friday before the arrival of the Imam while people listen to him, and when the Imam arrives, that person stops recitation. The original rule concerning the acts of 'Ibadah (worship) is that they are Tawqifiyyah (bound by a religious text and not amenable to personal opinion). The Prophet (peace be upon him) said,
If a person does any act for which there is no sanction from our behalf, that is to be rejected.
Narrated by Muslim in his Sahih (authentic) Book of Hadith.
May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.