Our boys grew up there. Never had to worry once you entered those gates. My wife once left her purse in France at Epcot and didn't realize it (we were carrying a backpack too) until we were in Mexico. We left the pavillion and she had a look of panic on her face. Within seconds, two suits (Disney people not in uniform) approached us to find out what was wrong; they were reading expressions at this point. Long story short, her purse with cash, credit cards and travellers checks were turned in by a guest and safe.
I know it doesn't sound like much, but that's typical of how things work in the parks. The happiest place on earth!
Nick & Lorraine