That's a bit of a leap. One could safely say that one person in the police department revealed it, but it goes against typical procedure to suggest that those investigating this crime were, at the scene, forthcoming to all the witnesses and circle of friends, which frequently includes the perpetrator, and would disclose details about the manner of death.
Also, since the "What do you think, she ******* bled to death." was said on the afternoon of November 2, she hadn't read it in a UK tabloid that would not published until the following day.
There were a couple of instances of "how could she know that if she were not there during the crime?" in her trial testimony. In her testimony about why did she tell Meredith's English friends that Meredith was killed in the closet, after lengthy persistence of Mignini and the court to have her answer a simple question she finally conceded that her only source of information on the throat-cutting was RS. And the location of these conversations was a car--I assume with Luca and Paola, since they were the ones that gave AK and RS a ride to the Questura and were the recipients of RS' interrogation about what they had told the police, and what they saw at the scene.
I am not aware of any source of information that claims that any of the English friends, or the Italian roommates or their boyfriends--much less RS or AK--were told by the ILE that Meredith's throat had been cut.