Now, the fun part is that not too long ago I actually joined that big scary company that could have been a competitor to start pushing internal startup opportunities for new revenue models and products i was considering. I reached SVP level execs, and saw how political winds above the VP level blew whatever meaningful thing got in their way without recourse. Compared to running with my own startup, that type of intrapreneurship felt like walking through a tar-pit. If it requires political approval and funding your "enemy" PM will likely have to fight a harder battle than you.
Make your product as good as it gets and you win, regardless of how many competitors know about it.
Look at the customer list of basecamp, any of their large enterprise customers could have built their own pm service, they didn't. Look at salesforce, how many companies could have built their own contact management system, they didn't. Wordpress, yammer, the list goes on and on. As long as you've got the right business model, you'll be fine.