I think the cases in which you need to keep something secret are pretty rare. I would also add that if this is your first startup you should get feedback as early as possible and validate your idea.
See if people are interested in what you are doing, if you are solving a real problem, and if people are willing to pay for your solution. It's easy to spend a lot of time building something you think people want, just to find out nobody cares, and you've wasted a lot of money.
You're going to need users. Why not start early? Start blogging, let people know about what you are doing, get feedback, and let people follow your progress. Then when you're ready to launch, at least you already have an audience.
Don't fear competition. Having competition means that there's money there. Most common argument I've heard about keeping the idea secret has to do with the belief that you have uncovered an unserved niche. Ask yourself why is nobody serving this niche? Is there any money there? That may explain the lack of competition. Also the lack of competition implies that you're going to have to educate your users, convince them they need your solution.
Ideas are almost worthless. Being first in the game for a viral idea is an advantage. You might reep the most benefits. But you also have to execute well and keep up your game. There's gonna be competition.
But for most ideas that solve a real problem, it's all about execution, and the way you solve the problem.
So I would say yes, go spread the word around, get feedback, get people excited about what you are doing and build an audience. Eventually you're going to have to tell people what you are doing. Competition will be there regardless, if it's a good market.