You still pay the full amount. Contributions to Social Security are an Individual thing.
You didn't give any details about what your work entails, but if you don't have any deductible job expenses, it will cost you more to be an Independent Contractor (1099) than an employee (W-2). You also may not have Worker's Compensation, Disability, or Unemployment Insurance.
As a side note, in most cases your employer can't legally be offering you that choice. The law dictates whether you are an Independent Contractor (1099) versus an employee (W-2). You can't legally pick-and-choose.
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