Obviously the original speakers of these languages were (the Hittites, etc.), but is this still the case today, at least for a significant number of speakers?
Or, more simply, how likely is it that I, a native speaker of English who is ethnically Germanic, am descended from the speakers of Proto-Indo-European who lived around the modern Caucasus?
It is statistically certain that you have ancestors who spoke Proto-Indo-European, as they lived over six thousand years ago. Even with relatively conservative estimates of the levels of interbreeding between ethnicities, your ancestors from that period of history would have been distributed throughout the globe. According to models described in this Nature article, you share exactly the same set of ancestors from that period of history with everyone else on Earth.
This is still an area of uncertainty, but one theory based on recent genetic evidence has the Yamnaya bringing proto-IE into Europe from the Russian steppes about 4500 years ago: