The common experiences attributed to gravity are simply the effects of a curved spacetime. Spacetime curvature can persist and propagate even in regions far from any mass or matter.
What mass (or more generally, energy density, momentum density, and stress) does is change how curvature at one time is related to the curvature before and nearby to be different than it otherwise would be.
An analogy would be electromagnatism. Electromagnetic fields can exist, persist, and propagate far from any charges, but charges allow the relationships between fields at different points in spacetime to be different than would otherwise be allowed. In fact, based on the relationships between the fields we can find out what net charges are around.
Similarly for curvature, give the metric, we can compute the Einstein tensor based on the changes in the metric (in space and time). From that we can find out whether there is net energy, momentum or stress in a region.
OK. So that's how mass and matter is related to spacetime (they can cause energy density, momentum density, and stress). But if you are far from any such sources, you don't know what (if anything) is causing that curvature, because that curvature is simply caused by earlier and nearby curvature.
Going back to our analogy with electromagnetism, it's like if you saw an electromagnetic wave zip by. It probably was caused by some charges in the past. But the equations themselves can't tell you that, if it has just been zipping around the universe forever, then it will continue until some charge or current somewhere makes it propagate different.
So when you notice a curved spacetime you can compute the metric and then the Einstein tensor and see whether it is zero, or whether it is electrovacuum or whether it is something else not corresponding to mass or matter. But that doesn't tell you whether there is mass or matter elsewhere else in the past that is responsible in some way for what you see in the here and now. There might be, there might not be. If what you see is an electromagnetic wave, then the analogy is strong in this example, and it cold have been caused by some stress, momentum, or energy in the past, or maybe it's always been zipping around. And looking at the curvature out here in the now doesn't and won't tell us that.
An extreme case of ignorance would be a stationary, static, spherically symmetric spacetime in a shell like region of vacuum. We can't tell from the solution in the shell whether it is from a black hole or from some matter. In the case of a black hole, we don't know what is on the inside, only the curvature on the outside. And something left on the outside in the vacuum must be the kind of curvature that propagates itself through the vacuum. Which is the kind that doesn't need mass or matter (to propagate or persist, though maybe it interacted with mass or matter in the past or elsewhere)
There was a research program called "mass without mass" by Wheeler that looked at solutions that from far away and long periods of time looked like there was some mass or matter in there, but didn't have any.