The first thing most people do when they are starting out on a new venture, idea or project is to tell the whole world about it. It's not bragging or showing off. They are genuinely excited about the whole thing and wants to share with the rest of us.
I think talking and discussing about your new idea with as many people as possible is the right approach. You want to find out everything you can about it and rarely anyone is going to steal your idea. It comes down to your execution anyway.
But there are some advantages to staying low profile when you are starting out. First is that it gives the ability to fail without being noticed. You can try a million different approaches in obscurity without worrying about what your neighbors think of you. To the rest of the world, it is as if all those failures had never happened.
When no one knows about you, you can make mistakes quietly. Learn at your own pace. Fail without the fear of failure. Failing in obscurity helps protect your ego and you’ll need your ego later on when you are successful. - Embrace Obscurity from 37signals
The other reason to work on the sidelines is to allow you to improve it until it's ready for the limelight. You get to decide when to reveal yourself to the world and your competitors will have less time to react. It's hard to prepare for something when you don't know what's coming.
New early-stage start up trend: get big quietly, so you don’t tip off potential competitors. - @cdixon
Businesses and ideas that appears to be overnight successes aren't what they appear to be. Most of them been through countless iterations and difficult times before anyone noticed them.
So, if you aren't famous or popular yet, that's fine. Work on that idea first. Being in the spotlight can wait.