I’ve read that the societies of the Middle Ages lacked the advanced and extensive bureaucracy of the Roman Empire. Would this bureaucracy have been staffed with commoners or middle-class individuals? If so, did this lead to any shift of influence between classes?

The Roman bureaucracy was interesting, from a class perspective. 

On the one hand, the Roman state relied heavily on local elites to collect taxes, pay for public works, events, and services, conduct imperial religious ceremonies, and going upward from those elites you got the imperial nobility who were often called on to become prefects, procurators, and other provincial officials, all the way up to imperial governors and consuls and the like. 

On the other hand, very low-status individuals, like freedmen, eunuchs, and slaves, could become incredibly high-ranking members of the civil service through proximity to the Emperor. So on an individual level, there could be quite startling social mobility (to say nothing of the more than a few emperors who started from incredibly lowly backgrounds). 

So in terms of the class effects on Mediveal Europe, I think you can look at what happened when the bureaucracy emerged in Early Modern Europe: it’s not that the nobility disappeared and the bourgeoisie took over, but there were more bourgeoisie in government than had been the case earlier, and a lot of them became noblemen themselves (see the “noblesse de robe”).