Another perspective that may help here comes from Bayesian statistics. You can model the outcome of competition between any two players using an unknown strength variable (say , $S_i$ ). Then, define:
Prob($i$ wins over $k$) = Prob($S_i$ > $S_k$).
You can use the logistic/normal distribution to model the above probability (like how the wiki page for ELO mentions. See the link provided by Joel).
As you pointed out, in the degenerate case, estimation of the strengths among all players in the set $A$ and all the players in the set $B$ would be possible but the estimates would not be comparable between a player from $A$ and a player from $B$.
The above suggests that we have to introduce some 'connections' between the sets $A$ and $B$ that would be informative about the outcomes between two players from these sets using either statistical or context-specific approaches to make the scores comparable. A few thoughts along those lines (one could use one or more of the following approaches):
Fix the mean and scale of the strengths to the same value for both groups.
Both the mean and the scale are not identifiable when you have relative ordering.
Use Latent Class segmentation approaches along with information about the competitive context.
For example, in the context of chess, we could classify players as 'Amateurs', 'Master', 'Grand Master' etc. We then impose some conditions on the outcomes of competition between any two players from these 3 classes. As an example, a 'Grand Master' will defeat an 'Amateur' 99% of the time or perhaps in 25 moves or less and so on.
Bringing context-specific information such as the above would enrich the data we have which would help makes the scores comparable.
Use Hierarchical Bayesian approaches to stabilize estimates with low amounts of data.
If players (between the cohorts $A$ and $B$) have not played that often hierarchical estimation would help stabilize the estimate of their relative strengths.
You could then investigate the extent to which you have to introduce 'connections' between the two sets to get a meaningful comparison. You would still need to define what you mean by meaningful comparison in your context. (Perhaps, define a meaningful comparison as one where you are able to correctly predict the outcome of competition between two players 95% of the time?)