To start with, I am going to assume that the child is the biological child of both of you and that you live together. In that case, only one person can claim all the tax benefits -- dependent exemption, child tax credit, head of household filing status (sometimes), EIC (sometimes), and dependent care credit or tax-free FSA.
The Statute IRC 501(c)(4) provides for exemption of: Civic leagues or organizations not organized for profit but operated. exclusively for the promotion of social welfare. Local associations of employees, the membership of which is limited to the. employees of a designated person or persons in a particular municipality.
Private Non-Profits. Private non-profit organizations receive financial support from only a few sources, such as a dedicated donor or investment income. In some cases, a private non-profit organization may receive a large portion of its funding from the individual who founded the organization. Founders of private non-profit organizations may secure additional funding from other wealthy donors who are interested in furthering the same cause.