Mattel didn't call it the "world's first interactive doll", they called it the "very first fashion doll that has continuous learning". The key words here being "fashion doll", which allows them to differentiate themselves from competitors they don't consider fashionable, and "continuous learning", i.e. the server is keeping a log of previous conversations, and using them to have better conversations.
The Cayla doll for instance was brilliant at answering questions like "What is Polytetrafluoroethylene?", and rubbish at answering questions like "Do you like kittens?" Sadly, it was a rather dull toy for a 7yr old girl, despite looking initially exciting.
What it didn't do, and this Barbie won't be doing, is snooping on us. I'd expect it to use Bluetooth to communicate with an App, which will then be using WiFi to connect to the Mattel server. If you don't have the App turned on, and by the sounds of it, if you don't press the button on the doll, it won't be listening. So if a child is playing with it, don't discuss national secrets, and if a child is not playing with it, there's no snooping.
The online conversation history is interesting, but not worth getting worked up about. It's recording how you've played with it, like all sorts of other online games keep logs.
Sure, there's a potential for abuse - "Hey - you say you like ponies! Did you know Mattel make a Barbie pony that's only $9.99 from your local Wal-Mart?" but I think if they start doing that, the backlash of people not buying their products out of sheer disgust will stop them pretty quickly. I'd say the worst is probably more like "Hey - you say you like ponies! I've got a pony called Muffin" without mentioning that Muffin costs $9.99 from Wal-Mart (or might do if it existed).