Nanny employers who are sole proprietors and have other business employees may chose to include their nanny's payroll in their 941 or 943 tax returns. The 'nanny taxes' are exactly the same employment taxes paid for other employees and do not require special calculation.
What the sole proprietor may not do is include the nanny's payroll and associated payroll taxes in their business tax deductions. A nanny provides childcare and is a contributing member of a household, not the employer's business.
Sole Proprietors who do not wish to co-mingle their household and business payroll may choose to report the nanny wages on Form 1040 Schedule H and would then remit the W-2 form at year end with a separate W-3 cover form.