Reference: Fortunato, Azar, Piot, Menick, Hessel, Osband, Graves, Mnih, Munos, Hassabis, Pietquin, Blundell, and Legg, "Noisy Networks for Exploration", ICLR, 2018 (arXiv:1706.10295), equations 9-11.

Factorised noise, which is the variant the paper uses for the single-threaded agents: rather than one sample per weight, it draws $p$ for the inputs and $q$ for the outputs and forms

$$\varepsilon^w_{i,j} = f(\varepsilon_i) f(\varepsilon_j), \qquad \varepsilon^b_j = f(\varepsilon_j),$$

so a layer costs $p + q$ random numbers instead of $pq$. The paper's stated reason is compute time, and it is the reason to prefer this form here too.

The two noise vectors are arguments rather than always drawn, because a sample has to be held fixed across a whole forward pass — and, for an agent, often across a whole episode. Drawing inside the function would make every call a different network.

Gradients reach $\sigma$ through the noise, which is the point: the scale of the exploration is learned rather than annealed.