How would a private caregiver file, especially when their employer did not take out medicare, federal, or social tax. And only provided a 1099-NEC.

You are being considered an independent contractor as opposed to an employee. There is a question as to whether you should be considered an employee and I have given you a link below which discusses that issue. Since you have to consider yourself a business you have to complete Sch C and unfortunately be subject to self employment tax. 
https://www.care.com/homepay/1099-vs-employee-why-the-difference-matters-when-you-hire-a-caregiver-1212110431