Can parties just agree among themselves about child custody without the need for “mediation?”

Yes. Parties often reach a settlement themselves or with the help of their attorneys. Less than 10% of divorce or custody cases ever go to trial.

As an aside, there are official training programs for mediators.  A person does not need to be an attorney to become a mediator; however, a mediator has to have taken the official training program or get an exemption from that requirement to appear on a court list of available mediators.

It is not “illegal” for someone who is not a mediator and who did not have “official training” to help a couple resolve their differences.  Sometimes this is a friend or family member.  Sometimes this is a member of the clergy or a marriage counselor.  The court is supportive of any respectful and effective means of resolving issues.