In a worst case scenario, I've seen PMs at a large company that was building an internal tool for its own use watch pitches by startups who ran white-label products with the same functionality. They did it to see if there was a better solution and if there wasn't they still got ideas on things that maybe should be included in their product. In the end, execution and scarce internal resources meant the final internal product was far worse than market options, so I wouldn't worry about that.
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.