The true fallacy of thinking someone is going to steal "your" idea is in thinking the idea is uniquely yours in the first place, especially if the idea is more of a concept or an entire product rather than something very small like a new feature.
Smartphones were thought up long before the iPhone, web search was thought up long before Google existed, personal tablets were thought up long before the iPad existed, the internet/web itself was basically invented in fiction long before it became a real thing.
I've lost track of the amount of times I've had some grand idea that nobody was doing only to see an announcement months/years later from company XYZ whose product was pretty much exactly my idea, even though I've never publicly shared the idea.
Product/service ideas are always, to some degree, just remixes of older ideas allowed for by current technology. When you have your insight, there are certainly dozens/hundreds of other people around the world with backgrounds similar to your own having vaguely the same idea at virtually the same time. Execution (or in most cases, even bothering to turn the idea into reality) and timing are everything.
Having said all of that, being in stealth mode still isn't always a bad thing because sometimes knowing how you are executing on an idea can be helpful to other people already on the same path as you are.