In the Roman Republic were debts and loans taken into account by the Censors when calculating the worth of a citizen's property? If yes, how were these checked?
Yes, it appears that debts were taken into account by the censors, and that details were provided as part of the statement given by every Roman citizen to the censors.
This is an interesting question, and is complicated somewhat by the nature of Roman Law.
Every Roman citizen was required to provide a statement of his familia (including his wife, children, slaves and freemen if he had them) and his property. The property which had to be reported to the censors was limited to that held under Quiitarian Ownership. this would include land (which was considered to be by far the most important asset), slaves, cattle, clothing, jewels and carriages (Livy, History of Rome, xxxix. 44).
This was important, as a persons value in the census determined their position in society. The census senatorius, would be the estate of a senator; while the census equestris, was the estate of an eques (sometimes translated as 'knight').
The punishment for providing a false statement was to be flogged and sold into slavery. His property would also be forfeit.
Based on another passage in Livy (History of Rome, vi. 27):
The year required censors also, chiefly on account of the uncertain representations regarding the debt; the tribunes of the commons exaggerating the amount of it on account of the odium of the thing, whilst it was underrated by those whose interest it was that the difficulty of procuring should appear to depend rather on [the want of] integrity, than on the ability of the debtors.
it seems the census did indeed show the amount of a man's debts and the names of his creditors. Further, it is clear that there could also be disputes over the value of that debt.
There is a reasonably accessible description of the Roman census in Charles Knight's Political Dictionary of 1845 (beginning at page 464).