Dear Nick -- First of all, if a ring satisfies Serre's criterion $S1$ and is "generically reduced", i.e., the stalk at every generic point is a field, then the ring is reduced. This is explained, for instance at the top of p. 183, Section 23 of Matsumura's "Commutative Ring Theory". Second, if $Y$, resp. $Z$ is a closed subscheme of a regular, locally Noetherian scheme which is itself regular, then it is everywhere locally cut out by a regular sequence, cf. Theorem 21.2(ii), p. 171, of Matsumura. Finally, if also $Y\cap Z$ has the "expected codimension", then $Y\cap Z$ is also locally cut out by a regular sequence, and thus Cohen-Macaulay, by Theorem 17.4, p. 135 of Matsumura. (In fact it is even LCI by Theorem 21.2 again.) A Cohen-Macaulay scheme satisfies Serre's criterion $Sn$ for every integer $n\geq 0$. Thus your scheme $Y\cap Z$ is reduced.