From what I see at https://www.irs.gov/taxtopics/tc756 it states "Household workers are your employees if you can control not only the work they do but also how they do it."
But from what I see at https://www.irs.gov/pub/irs-pdf/p926.pdf, I interpret it in a way that makes the nanny a private contractor. By my interpretation (I am no tax expert by any stretch) the fact that the work is flat out not performed in your home or any any home or other property that you own, rent, or have other control of, disqualifies them for the household employee status.
But that's my two cents.
Based on that, you'll need to issue them a 1099-MISC. It is important that you have their taxpayer identification number before you pay them anything. It can be their SSN or their EIN - it doesn't matter. So have them fill out a provide you with IRS Form W-9 (https://www.irs.gov/pub/irs-pdf/fw9.pdf) before you pay them anything. If they refuse to provide the information, then they just flat out don't get paid - it really is that simple.
Now I'm sure you can understand a person's reluctance to provide their SSN. That's not an issue. Anybody can get an EIN in about 10 minutes and it's absolutely free at https://www.irs.gov/businesses/small-businesses-self-employed/apply-for-an-employer-identification-n... So it may just be a matter of you informing them (the nanny) of this. Note that you flat out can not claim any childcare expenses without the childcare provider's SSN or EIN.
