There's a lot of discussion about the security of Apple's iPhone NameDrop feature. Police departments are recommending turning this off. It's gone viral, and given Apple a bit of a security/privacy black eye.
It's also overblown. Key things to know:
1. If your phone is locked (say, at the gym) it will not share contact info via NameDrop.
2. NameDrop only works when the phones are unlocked and the phones are physically in contact with each other. Just being "close" isn't good enough. Nobody is going to swipe your data by walking past you.
3. Your phone will ask you if you want to share data. It won't share automatically.
There's more detail on this here. Here's how you turn NameDrop off if you want to:
Open Settings, then select General. Select AirDrop and disable "Bringing Devices Together" and "Use Cellular Data". You're good to go.
That said, you can share contact info via text message with anyone you want to, so this feature seems pretty useless to me. I have disabled it on my phone, and I strongly recommend you disable it on any child's phone (or iWatch).