Because Neanderthals lived mainly in Europe and Asia, a signal for introgression would be matching regions of DNA in Neanderthals and Eurasians that are not found in sub-Saharan Africans. While one commonly thinks of Neanderthals as primitive, they almost certainly had useful adaptations not already present in the small gene pool of modern humans, newly arrived from Africa. (Neanderthals had been in Europe and Asia for roughly 300k years before modern humans.) Adaptive or selectively neutral Neanderthal genes are more likely to have introgressed into humans than non-adaptive ones (see below).
2009 interview: ... We are analyzing this and it's a very difficult analysis, because if there are tiny parts of a genome that were contributed to humans, you have to assume that they are not due to some error in our analysis, or contamination of the DNA of modern humans, or any little bias in the algorithms we use.
In the public media we're pretty much depicted as saying there was absolutely no mixture. It's very hard to convey these subtle messages to the public. If you read our papers, we say very carefully that there is absolute proof that they didn't contribute mitochondrial DNA. That doesn't mean that they couldn't have contributed other parts of their genome.
... It's very clear from the genetic evidence that the big picture is the out of Africa one. That's not to say that there couldn't have been a small contribution from earlier archaic forms such as Neanderthals in Europe to present-day Europeans or erectus forms in Asia to Asians, but it has to be very small. ...
But the Neanderthal genome will be a chance to address that. And it depends on what you're interested in. As a geneticist I'm not so interested with who had sex with whom 30,000 years ago. The question is, as a geneticist today: did Neanderthals contribute significantly to our gene pool today? Did the Neanderthals have an impact on the variance we carry? And that's got to be small.
For a plausible argument for introgression of genes from Neanderthals to humans, see The 10,000 Year Explosion, by G. Cochran and H. Harpending:
... Logically, if admixture occurred at all, it had to happen somewhere in Neanderthal-occupied territory, which means Europe and western Asia. As modern humans expanded their territory, they must have encountered Neanderthal bands again and again. The two kinds of humans coexisted for a few thousand years before the Neanderthals disappeared, at least in some regions. This looks to be the case for the Châtelperronian culture of France and northern Spain, and there are traces of a similar culture in Italy. If there was trade, or if there was enough contact to transmit toolmaking techniques, there was sexual contact as well—depend on it. If in the future we look at very large genetic datasets from huge numbers of individuals, we might ﬁnd a few traces of neutral Neanderthal genes.14
... Imagine that humans occasionally mated with Neanderthals, and that at least some of their offspring were incorporated into human populations. That process would have introduced new gene variants, new alleles, into the human population. Many, probably most, of those alleles would have done almost exactly the same thing as their equivalents in modern out-of-Africa humans; they would have been neither better nor worse than those equivalents—in other words, they would have been selectively neutral. Those neutral alleles from Neanderthals would have been rare, and they would probably have disappeared, the typical fate of rare neutral alleles.
... J. B. S. Haldane, the great British geneticist (1892–1964), found a systematic way of adding up all these probabilities, and his method yields a surprisingly simple answer. If the allele confers an advantage s, its chance of going all the way is 2s. In a stable population, a single copy of an allele with a 10 percent ﬁtness advantage has a 20 percent chance of eventually becoming universal. The fate of one copy of a favorable allele is very much like that of a gambler who starts out with one chip and a roulette system—a way of beating the odds—that really works. ...