You won't be able to raise money from experienced investors if your intellectual property is disputed or could potentially be disputed. When you do go to raise money, you'll have to make a number of representations about your company, and one of those representations will be that all the IP belongs to the company. (As an aside, you'll also have a hard time raising money if you're still at your full-time job, unless your app has some serious traction. Investors like to see commitment.)

When we were getting started, my cofounder and I held off on putting a single word to paper until we were out of our full-time jobs, on the advice of our lawyers.

I'd just go and talk with your employers about it. "Hey, I'm really happy here, but I've been thinking about a side project - I wanted to arrange with HR that it was my own invention and belonged to me, of course it won't interfere with my job which is my first priority..." Employers want to keep good people so if your side project is truly unrelated to your current company it hopefully won't be a problem.