It seems like Twitter recently changed some of the authorization structure and applications seem to default to read only now. I think some of the instructions for setting up the ability to post from WP to Twitter might need some updates. https://dev.twitter.com/docs/application-permission-model
I'm having trouble getting the settings all right. I think I've got them figured, but haven't gotten it to successfully post from our latest WP site just yet. Older setups work fine.