In California, if someone is claiming the litigation / judicial proceeding privilege, who has the burden of proof?

The person claiming the privilege is usually the one who has to prove it. And as to this issue, the answer should be the same across the state. There also are limitations to privileges that apply in certain circumstances.

To narrowly answer the question you asked above, the one claiming the privilege has the burden of proving that such privilege exists, as another attorney stated above. In this context, X would have to prove that such privilege exists by filing an "anti-SLAPP" motion to strike Z's defamation complaint. SLAPPs are "Strategic Lawsuits against Public Participation." In short, they are lawsuits brought to stop individuals from exercising their rights to access the court system for things like defamation actions. In an attempt to protect the rights of free speech and public participation, California passed Code of Civil Procedure section 425.16 and other supplemental laws, colloquially known as "anti-SLAPP" laws. That said, the chances of an anti-SLAPP motion being successful turn on the specific facts of each case, so it's very difficult to answer in general terms. Because what constitutes "public participation" is construed very liberally by the courts, if the words were necessary and relevant to prove the merits of X's underlying case against Y, an anti-SLAPP might very well have a good chance of being successful. Conversely, if Z could prove that X's statement was not relevant or necessary to X's claim against Y, then the litigation privilege would likely not apply, and the anti-SLAPP motion would not be successful.

That really depends on where the statement was allegedly made. If the statement was made in open (public) court, then there is no privilege that exists. What you say in court, is usually fair game and open to the public (and often televised and/or reported in newspapers, etc.). Just because you're in a lawsuit doesn't make the statement privileged. In fact, if there is a transcript of the statement, that can be used in other judicial proceedings. For example, if you testify in court that you were speeding and caused a car accident in a criminal proceeding, that can often be used against you in a civil case to prove negligence. So I think there is a presumption that there is no privilege and she would have to prove why it is. The other missing fact is, in what context are you trying to get her to admit to these alleged statements? If you're asking in a casual conversation, there is no right to admit to anything. If you are cross-examining her, then she may be compelled by the court to answer truthfully. Short answer - without knowing how/why/where you're trying to get her to admit these statements and where/when these alleged statements were made, it's impossible to answer your question.

Typically the person asserting it has the burden, but Google it.