In England we 'sit' or 'take' exams. We often take several exams in the same period of time. For example, we might sit a science theory exam on the Monday and the practical exam on the Tuesday. There may be other tests we take that relate to the main exams. These would probably be taken (sat) close in time to the main exams. I think the problem is the use of the word 'sit' ! Alongside can mean "side by side/next to" (position/place) or it can mean 'simultaneously' (time!). So 'sit alongside' could mean people or things are close to each other in position, or it could mean exams taken in conjunction with each other (therefore probably very close together in time!) Usually we say "Alongside OF..." when we mean together in time/in conjunction with. Well, I do.

In this instance, "sit" means to take the exam. In British usage, one "sits for an exam"; in the US, one "takes an exam". The sentence means that the students take the common tests at the same time that they take the final exams. Not necessarily at exactly the same time, and not necessarily simultaneously, but certainly within a short period of time.