It is passed down from the earlier scriptural writings, when women would go to the mikvah before marriage, after childbirth, etc. Many would take their friends and female family members, but that was just for friendship and companionship. Then, with regard to men who were engaged, and for other reasons to go to the mikvah, they could go alone or in a group of chosen males.
One also went to the mikvah upon taking on a different teacher, changing their name, after a long illness, after an important understanding of Scripture was fully accepted, etc.
Often, someone would accompany the one going in, in order to assure that the person dunked completely under, to the tips of the hair, but that was not commanded; it was, however, convenient. The command was not to have someone present but to just do it.
And the person who went into the mikvah said the words, not someone else. There was no one there to push them under; they dunked themselves.
Some churches like to make a difference between going to the mikvah and being baptized, but there is no difference, other than church tradition that has been added.
