In English, I believe the notation $p(x, z \mid \theta)$ is pronounced something like;
pee-of-ex-zee-given-theta
But I sometimes see a semicolon used, as in $p(x, z \,; \theta)$, which I believe indicates that $\theta$ is a considered as a parameter of the model $p$, not a random variable.
Mathematical interpretation aside, how is the semicolon ; in this expression pronounced?
I don't think there is community-wide consensus on the question of pronunciation, but you are right, the semi-colon is usually meant to indicate that the expression depends on $\theta$, a fixed (non-random) parameter. I would personally pronounce it as "given-theta" or "given-fixed-theta".
You could also write $p_{\theta}(x, z)$, which would be pronounced "pee-sub-theta-of-ex-zee".
References on the meaning of the semi-colon: What does a semicolon denote in the context of probability and statistics? and Meaning of probability notations