If the fields $b$ and $c$ have conformal weight $\lambda$ and $1-\lambda$ and action is:
$$S = \frac{1}{2\pi} \int d^2z \, b \bar \partial c,$$
under conformal transformations $z \rightarrow z+\epsilon(z)$, we can get the energy-momentum tensor: $$ T(z) = - \lambda :b \partial c: + (1-\lambda) :(\partial b) c:. $$ But this symmetry $z \rightarrow z+\epsilon(z)$ is local, shouldn't this be the energy-momentum tensor for global translations: $z+ a$? I am getting confused as to why a local symmetry is giving us a stress-energy tensor.