Twitter has both public and private messages. If anyone is taking about monitoring communications on it, I would assume they're talking about the private ones.
Why would you assume that ? - a large amount of the "incitement" that took place caused problems in a large part because it was public in a one-to-many form.
I'm sorry, did I miss the part where they said they would only monitor Twitter? Or am I remembering correctly where they used the broad, sweeping term "social networks", which would include Facebook, and I'm sure BBM chat.
Do you live in a fantastical world where governments pass very strictly limited, narrowly defined laws that precisely target one specific problem? Because here on earth the rule is overbroad laws made by ignorant legislators.
> I don't really see how anyone could object to a government monitoring it
Because of small thing called 4th amendment which in part reads "The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated"
While a USA constitutional amendment is based in British law. And in case you question; yes, I consider my electronic communications/records to be "papers, and effects". And if you question the wisdom of that particular right, I don't have time to explain, reference history for the supporting evidence.