Yeah, that's the Out of Africa theory.
One common ancestor and all that. Not very likely though.
I guess the fact that there were denisovans in Spain 100.000 years before denisovans and neanderthal supposedly left Africa kind of busts that myth wide open.
Perhaps, that is why some ethnicities have much greater denisovan heritage (asians) and neanderthal heritage (whites)?
The reason they think we came out of Africa is to do with our X and Y chromosomes (the bit of DNA that gives us our gender, which is in the nucleus of our cells).
They change very very slowly (on evidence only every 10,000 years, and when they change, they develop
additional markers, while retaining the original markers).
If you think of the mitrochondrial (X chromosome) DNA as an enormous string of letters, every single human on earth has the exact same marker in their mitochrondria at position 4312, which we don't share with other homonids and apes.
Different groups have
additional markers. So people from southern India will have the african marker at 4312, the east african marker at position 16223, the asian marker at position 10400 (known as group M), and some further markers indicating different indian ethnicities. Someone from Europe will have the african marker at position 4312, the east african marker at position 16223, and the indo-european/caucasian marker at position 12705 (known as group R) with further additional markers denoting different sub-branches of europeans.
The key thing is that we all have the same african marker at 4312, and everyone outside africa has the east african marker at position 16223 as well, which is why they think that the continents outside Africa were populated by a small group of east africans who left by crossing into arabia. (Carbon dating of skeletons indicate they left Africa thousands of years after the Neanderthals did).
As to "why africa and why east africa" - just because the oldest skeletons with these markers have been found in Africa. If the oldest skeleton with these markers had been found in Europe, then we'd be all "out of Europe". However, the very old homonoids found in Europe don't have this common marker at position 4312 on the mitochondria. And the oldest skeletons with human Y chromosomes are also from Africa.
Mitochondria (the X chromosome) is passed from the mother to all her children, but only the daughter passes it on to her children (so all girls have XX and boys have XY).
If a homo sapian female had mated with a neanderthal male, all her children would have the homo sapian mitochondria, with the tell-tale marker at position 4312. If a neanderthal female had mated with a homo sapian male, then her children would have the neanderthal mitochondria but her male children would have a human Y chromosome. If a neanderthal female mated with a neanderthal male, both the mitochondria and Y chromosome would be neanderthal, and if a homo sapian female mated with a homo sapian male, then both the mitochondria and Y chromosome would be human.
These are the only possibilities - it's not possible to be descended from a neanderthal and not show either a sign of a neanderthal female ancestor or neanderthal male ancestor through the X and Y chromosomes.
So in order to prove that some people have either a male or female neanderthal ancestor, you need to find someone without those human gender markers at position 4312 on the mitochondria- and so far they haven't found them.
Before the genome was sequenced, the Chinese believed that they were descended from neanderthals and this made them "different" and "special". (Archaeologists had found evidence of a huge neanderthal settlement in China complete with skeletons and tools).
But when their mitochondria was sequenced it had that tell-tale marker at position 4312 which the neanderthals didn't have, and which the oldest skeleton with it being in africa. And when Y chromosomes were sequenced, they too showed a common african homo sapian ancestor rather than a neanderthal ancestor. So they too have come out of Africa.
I suppose it's possible that somewhere on earth there is a descendant of a neanderthal - but they are testing DNA all over the place - modern people, ancient skeletons, remote tribes, people in cities, and they haven't yet found someone without that african marker at position 4312 on their mitochondria.
Perhaps it could be you! Go get your DNA tested.