It's actually quite hard to write meaningful tweets that could fool someone with a bit of knowledge of the organisation, such as a journalist who has had it in his portfolio for a while.
Plus, a journalist would have the contacts to verify the information. Traditionally, the government doesn't lie to American journalists – they may decline to comment, or give you a convoluted answer that doesn't actually answer your question. But they wouldn't lie. Considering these organisations are supposedly full of people sharing the leaker's point of view, it shouldn't be difficult to get confirmation at least off-the-record.
Of course all that requires a certain amount of trust in the media.
Plus, a journalist would have the contacts to verify the information. Traditionally, the government doesn't lie to American journalists – they may decline to comment, or give you a convoluted answer that doesn't actually answer your question. But they wouldn't lie. Considering these organisations are supposedly full of people sharing the leaker's point of view, it shouldn't be difficult to get confirmation at least off-the-record.
Of course all that requires a certain amount of trust in the media.