Where do you stand on the debate that proprietary software is more secure than open source software?

Neither proprietary nor open source has proven to be more secure than the other. Until we can write the perfect code for the perfect hardware, there will always be vulnerabilities. Since perfection is unachievable, so is perfect security. The best we can do is be vigilant in our monitoring, patch whenever possible, and plan for the worst. If you are prepaired for the inevitable, when it happens you won't be caught with your pants down and screaming "why me?".

Firmly against. Both are vulnerable, but at least open source has lots of eyeballs looking at it -- often expressly to see if they can find anything nefarious.

Neither one is more or less secure then the people securing it. It doesn't matter if you create the ultimate security program it will always have a vulnerability and that's people. Something as big as the DODIN might have better security protocols but is less secure because of how many people access it. A private firm might have less security protocols but be more secure because it's not as large of a target and doesn't have as many human vulnerabilities.