Both are common. I don't think there is a norm. (Although, professional, highly experienced nannies who aren't in the country illegally tend to be legit.) Some things to consider:
1) Paying under the table tends to be cheaper. You can pay them less/hour and they will still make more because they aren't paying taxes.
2) If you want to claim a childcare benefit on your taxes, you have to go legit. You'll need an employer ID number, etc.
3) If you're afraid of potential liability issues, go legit. You'll be covered under your state's plan. Of course, if your nanny is not legit, it's unlikely he/she'll go after you for lost compensation if she gets hurt on the job. But, any injury in your home might have to be paid out of your home insurance, and your insurance company might balk if they find out she's an under the table nanny.
4) If you or your partner get a job in the future with any sort of security clearance or political issues, this could come up and make it difficult.
So, while there are benefits (easier and cheaper in the short term) to paying a nanny under the table, there are potential consequences.
Good luck with your choice.