— Searching for a company on Google often resulted in stock quotes on that company for a day or two, even though I don't own any stocks.
— When friends and relatives travel together and share flight information over Gmail, Google Now can't figure out which one is yours. It was telling me to head to the airport in Taipei, Taiwan, where my brother had a connecting flight, even though I was still in Bangkok waiting to leave.
— When I'm near a Barnes & Noble, I often get a card saying I could research products there. Clicking on the card gets me a link to the retailer's website and a chance to "Scan for product information." There was a camera icon, so presumably I could photograph a bar code or other identifier. It's not clear why I'm limited to being at a Barnes & Noble to do that and why Google thinks I need it just because I'm near a store.
Despite getting stray information at times, I find Google Now useful enough to leave it on. I could always customize the service by telling it never to give me stock quotes, for instance. And some of the cards are enabled only when I have Google's Web History feature turned on through my Google account settings. (Newer accounts come with that feature already on, but you can turn it off and still use Google Now.)
You do have to give Google Now permission to scan contents of your Gmail account, but it's typically limited to confirmation notices from airlines and hotels rather than discussions of hobbies and medical conditions. You also must give it permission to access calendar entries. Privacy worries aside, Google Now's appeal is in what it does with that data. That's why I'm OK with Web History.
Whatever you think of the rivalry between Google and Apple, don't look at Google Now as a Siri-killer. Think of it as a companion for the tasks you can't accomplish with a simple voice search.