That's normal and, frankly, it doesn't matter. Tell it to your therapist, keep it out of your correspondence. It doesn't matter if you feel insecure if you can act and write like you're not. First of all, Hollywood itself is notoriously insecure so you're in good company. But that's what makes confidence on the part of a writer so intoxicating.
Confidence is a power and power attracts power.
Perhaps because they are more familiar with less attractive ones. You also have a natural insecurity if you aren't a Randall Wallace or Brian McQuarrie that can immediately command attention with their Oscar status. Most of you don't have proof that other professionals think you are a great writer, or even a good writer. But you don't need to have proof to get a script read. You have to believe that reading your script will be the proof they need.