So staying small and out of the public spotlight to iterate on failure is valuable (and doesn't produce any negative PR).
If you have to go open beta to get that feedback, then do so. If you can get away with a private beta, do so.
Here's the better question: Can you set up a system so that you don't even need any sort of beta to tell?
You probably want users to be able to freely refer your product to friends. You probably want people to be able to discover, sign up, and try your product in one sitting. You probably don't want potential users to lose enthusiasm while they wait for a beta invite. You probably won't have crushing server traffic. If you do, you probably won't regret letting so many people use your product at once.