If I pay and individual for childcare and use their SSN on form 2441, what do they need to do when filing their taxes in reference to that?

[Critter-3]

They must report the income usually on the Sch C. 

[AmeliesUncle]

The same thing as if you did not file Form 2441 with their information:  They just report all of their income on their tax return.

As Critter pointed out, if they are watching the child outside of your home, it is likely they are Self Employed.  But it depends on the facts-and-circumstances.  If they are watching the child in your home,  you should probably be treating them as a Household Employee, in which case you may want to consult with a professional about that situation.