Have there been enough studies to draw any definitive conclusions on how well sites such as Duolingo or Livemocha work in comparison to classroom learning?
Not restricting it to Duolingo et al. specifically the name of this research field is CALL (computer assisted language learning). There's at least two major journals devoted to this in all of its aspects:
ReCALL - which is the journal of the Journal of the European Association for Computer Assisted Language Learning
CALL - published by Taylor and Francis
Having done a brief amount of research in the CALL field, I would say that most researchers are well aware of the following pitfalls:
- Completion rates - even in academic courses for credit, less people complete online work.
- Motivation - this is linked to the above but most students find it very hard to stay motivated for online learning.
- Level of interaction / limitations for speaking practice
Separately, I would add material quality (at least with many of the resources for Japanese speakers to learn English).
Here is one such study, but it appears to be at least commissioned by Duolingo itself even if it was carried out independently. Here is an independent critique written by Michael Schmitz, an online German educator which compares Duolingo to his more traditional classroom courses.
In general I am of the opinion that there are not enough studies to make a definitive conclusion about overall effectiveness, mostly because the choice to use Duolingo to learn may largely depend on how one prefers to learn the language. Duolingo can be used much more casually than a classroom course so it is somewhat difficult to compare them in that respect. However, you should be able to find enough information to decide how you would prefer to learn (and if you're still not sure, you can always try out Duolingo because it's free).
Note: Duolingo even has a research page that you can visit to learn more and which might be updated with related studies in the future.