As a long-time liberal, my opinions of the mainstream press have long been that they are far too accommodating of clearly false conservative talking points and too willing to advance the conservative narrative without in an attempt to seem unbiased, especially with regard to the intersections of politics and science or human rights. The press were cheerleaders for Bush's invasions, for the equating of capitalism and freedom, and many other ills.
By contrast, my impression of the conservative complaints regarding the press is that when conservatives are not allowed to dominate the conversation, they feel oppressed.