Hmm that in interesting. I work for a public school district and on my W2 they do Box 1 Wage - Box 2 Federal Income Tax Withheld and Box 5 Medicare Wages - Box 6 Medicare withheld, but they don't do Social Security since we pay into our states Teacher Retirement Fund, which, on box 13 is marked as a Retirement Plan.
So using mine as an example, I can see then not withholding Social Security, since you are paying into a retirement plan, but not sure why they would not pull for Medicare.
Now from past experience, you are going to be responsible for a portion of anything the IRS say is owe.
"Both employer and employee hold the responsibility for collecting and remitting withholding taxes to the Internal Revenue Service (IRS). For the most part, the employer withholds these taxes on behalf of their employees, but in cases where an employer does not do this, or where an employee is self-employed, it is the responsibility of the employee to pay these withholding taxes."
https://www.irs.gov/compliance/criminal-investigation/employer-and-employee-responsibilities-employm...
