Suppose I have a singular projective variety $X\subset \mathbb{P}^n$ that is reducible with $X=\bigcup_i X_i$ smooth irreducible components. That is, the irreducible components are smooth but $X$ is singular along the intersections of the $X_i$.

I would like to compute the Segre classes $s_t(X,\mathbb{P}^n)$. In particular I (optimistically) wonder whether one can obtain these classes in terms of the Segre classes $s_j(X_i,\mathbb{P}^n)$ of the irreducible componentes. If not, what would I need more to compute $s_t(X,\mathbb{P}^n)$?