To what extent is the method of participating in prayers upon the Prophet, peace and blessings be upon him, legitimate, whereby a person distributes a specific number of prayers upon the Prophet, peace and blessings be upon him, to his acquaintances, then collects what they have prayed and adds it to the total of others?

The actions mentioned in the question—such as celebrating the Prophet’s birthday (Mawlid), specifying a certain number of prayers upon the Prophet (peace be upon him), and distributing schedules for that—are a misguided innovation (bid’ah dalalah) and were not done by any of the righteous predecessors (Salaf Sālih). Sending prayers upon the Prophet (peace be upon him) is a great act of worship, but it must be in accordance with the Sharia, not through innovation. Wasting time on such innovations is a squandering of life and a deviation in religion. It would have been more appropriate to utilize these efforts in teaching people the Prophet's guidance (hady) in their daily lives. These people have fallen into multiple innovations, including: making these prayers contingent on the innovated occasion of the Prophet’s birthday, specifying a certain number for these prayers which is not mentioned in the Sharia, and considering praying upon the Prophet a collective remembrance while it is a private remembrance. It is obligatory upon whoever invented this method to repent, and upon whoever knows about it to forbid people from it.