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It's legal to video in your own home in every state, and it's only illegal to record speech without a person's consent in the following 15 states: California, Connecticut, Delaware, Florida, Hawaii, Illinois, Louisiana, Maryland, Massachusetts, Montana, Nevada, New Hampshire, Oregon, Pennyslvania, and Washington.
So if you are outside those states, you don't have a problem. Inside those states you can check to see what your legal responsibilities are - for instance you may be able to record conversations you are having with someone, and given that you are guardian of your children you may be able to record conversations they have with people, which would make most recording completely legal. But if you're worried about it, simply disable or tape over the microphone. Honestly it's going to be hard for your nanny to prove you are recording their conversations unless you actually use the material against them in court.
I don't know why people are spending so much time telling you it's a bad idea. The best time to put in some sort of monitoring hardware is before you even suspect there's a problem. You have absolutely no obligation, ethically, morally, legally, or otherwise to tell someone that you are videotaping rooms inside your own home for the safety and security of your family. It's ludicrous to think that people believe their "personal privacy bubble" should extend into
your home when they are spending time alone with
your children. I simply don't understand the mindset around here.
Yes, ideally trust would be enough, but if your brain is sending signals that a situation needs to be
monitored (not even acted on - "Hey, let's just collect some information and make sure everything's going ok!") then, were I you, I'd pay attention to those gut feelings and do some monitoring. I wouldn't care about the legality of recording the audio because, at this point, I'm merely gathering information, and if I find out I was wrong, great! Everything goes in the bit bucket. If I find out there's a problem, then great! I can, ideally, resolve the issue long before anything goes too far.
But don't take the naysayers here to heart, and don't let them dissuade you from this path with the faulty idea that the privacy of the nanny is more important than the safety of your children and the security of your home.
However, I definitely agree with those saying that if you believe there is a problem, don't use the girlfriend. In this case it seems you merely don't have enough trust, but don't have any reason to distrust, so adding monitoring on top of the belief that you can trust those who tell you she's fine seems a prudent and reasonable action.
-Adam