For the Obama White House, tweets from reporters are a kind of early warning system. It’s up to Jessica Allen, 24, to sound the alarm. Allen, whose official title is “media monitor,” tracks journalists’ tweets and flags them in mass emails that land in the in-boxes of more than 80 Obama aides, including chief of staff Denis McDonough, White House counsel Kathy Ruemmler, press secretary Jay Carney and senior adviser Dan Pfeiffer. The result? Reporters who regularly cover Obama have become familiar with seemingly out-of-the-blue emails or telephone calls from officials taking issue with their tweets — often thoughtfully and...