For childcare if a relative is paid for taking care of my child while I work will they be responsible to file that income on their taxes

[Opus 17 · Accepted solution]

Yes.  When you provide the name and SSN of the care provider, the IRS will look to see if there is matching income on the other person's return.  Whether it is taxable or not depends on their other income and tax situations.  

Also, if care is provided in your home, they are your household employee, and you are required to issue them a W-2 and to withhold household employee tax and pay it to the IRS, unless the "relative" is your parent or a person under age 18.

https://www.irs.gov/taxtopics/tc756

If the person provides care in their home they will usually be considered self-employed and must file a self-employment return.  However, they can lower their taxable income by claiming some deductions for using part of their home for business purposes.