Nope, Blurtit never posts to any of your social profiles without permission.
The only way anyone on Facebook would see anything you've contributed on Blurtit is if you share it yourself using the social sharing buttons on the right hand side of the page:
The idea of asking people to sign in with Facebook or other social networks is to reduce the amount of spam we get on the site. While spammers usually have multiple email addresses, they often don't have as many Facebook accounts - which means we end up spending less of our time moderating, and more time making the site amazing :)