In 3+1 dimensions, the Ricci tensor vanishes when the stress-energy tensor vanishes indeed. Which means that, whenever there's a vacuum, the Ricci tensor vanishes as well.
But it also happens that, in that number of dimension, the metric is not entirely determined by the Ricci tensor. The full curvature tensor (the Riemann tensor) is a mixture of the Ricci tensor and the Weyl tensor. The Weyl tensor roughly describes how gravity propagates : if you have some energy, it will influence the metric of the vacuum regions around it.
I emphasize the 3+1 dimensions because that is not the case in 2+1 dimensions (gravity stops in a vacuum, because the Riemann tensor is $\propto$ the Ricci tensor) and even less so in 1+1 dimensions (the Einstein tensor is a topological invariant, and does not vary with the stress energy tensor).
In the case of Schwarzschild, what you have is a distribution of energy in the middle. There's several ways to describe it. You can use distributions, but Schwarze distributions handle non-linear equations poorly, and non-linear distributions are a rather thorny topic. But there's still some ressources talking about how to describe a singularity as a Dirac distribution. It has the benefit of being rather clear : The stress energy tensor is a Dirac, and so is the Ricci tensor.
You can show it indirectly, as well, by calculating the Komar mass. You find that, in an arbitrarily small region around the singularity, there's a mass M. The Ricci tensor is, as usual, 0 outside of the singularity and undefined at r = 0.
Of course you can also just have a plain old spherical body. In that case, the metric is just Schwarzschild outside (with energy density 0) and whatever may be inside, with, hopefully, a continuity in the metric between the inside and the outside.
A good way to illustrate how the curvature of space does not depend entirely on the Ricci tensor is to look at empty space. The simplest vacuum spacetime is Minkowski space, but it is not the only one. You can freely add sourceless gravitational radiations (using for instance pp-wave metrics), which will also be a vacuum solution.
In the case of the Schwarzschild spacetime, the stress energy tensor is not enough to get the full metric because you are not looking at all the spacetime. You are only looking at its vacuum regions without looking at its energy distributions, which can influence other regions. In the case of gravitational waves, it's a boundary value problem : solving the Einstein equation still requires, like all PDEs, some boundary value of the field.