Local and global definitions of pure infiniteness for a \cst-algebra $A$ are compared, and equivalence between them is obtained if the primitive ideal space of $A$ is Hausdorff and of finite dimension, if $A$ has real rank zero, or if $A$ is approximately divisible. Sufficient criteria are given for local pure infiniteness of tensor products. They yield that exact simple tensorially non-prime \cst-algebras are purely infinite if they have no semi-finite lower semi-continuous trace. One obtains that $A$ is isomorphic to $A\ot \Oinf$ if $A$ is (1-)purely infinite, separable, stable, nuclear and $\textrm{Prim}(A)$ is a Hausdorff space (not necessarily of finite dimension).