From my understanding (B.A. in Classical Civilizations and reading lots of books on Rome for fun), that slave traders would follow armies, or at the least show up after a victory. The Roman army or leader of the army would generally sell the slaves en masse to the slave traders at a discount cost. Then the slave merchants would be responsible for transporting and selling their new property elsewhere throughout the empire. So essentially the slave merchants would buy them at a discount, but since they bought so many the Roman army still made plenty of money to pay soldiers and line the pockets of the leaders of the army. This is my understanding, but if anyone has more information please correct me.

Letram is correct. Wanted to add a few things....

We only have sparse and incomplete information about the logistics of the transactions. Merchants could often follow Roman armies and buy up war prizes immediately as they became available. We also know that in other cases, slaves were moved to locations that were more appropriate for business transactions or even shipped to Rome, using the same military and commercial supply lines that kept the Legion afloat, to be auctioned off. We also have some knowledge of the Romans selling slaves off directly to local populations OR ransoming off their prizes to relatives.

I think an important detail is to add that slaves captured in war were not property of the state per se... They were the property of officers and generals or even individual soldiers. They would then either sell the slaves themselves or would sell them to traders. A recent Princeton paper on the subject had this little bit to say which may be of interest to you:

Considering the huge scale of the Roman slave trade, substantial amounts of capital must have been committed to the procurement and distribution of slaves, and large numbers of middlemen had to be involved in this business.

So, the trade kept afloat many merchants in the Republic. It was not however a considerable earner for the Roman state. Contemporary customs records from Palmyra had this to say:

Fiscal intervention probably only had a moderate impact on the volume of trade: tariff records from Palmyra from AD 137 stipulate customs dues equivalent to not more than 2 or 3 % of the value of teenage slaves, while the tariff recorded in an analogous inscription from Zarai in Numidia (AD 202) envisions an even lower rate. In Egypt, Roman authorities upheld the earlier practice of requiring export permissions and export fees (of unknown size) for slaves. (Believed to be 22 denarii for imports and 12 denarii for exports).

We may however say that even those light duties were likely to give the state massive amounts of coin. It was light in percentage of money taken from one transaction by the state but it is far more important to note the scale of the Roman slave trade and the reason the Romans did it... According the Schneidel,

slave societies’ are most likely to emerge in the context of relatively high real wages (i.e., demand for labor) and relatively low slave prices. It is plausible yet impossible to prove that Republican Italy conformed to this model.

Which means that the Roman economy may not have been what it was without its slave trade. Its easy to forget, in the scale of this question alone, that the Romans didn't simply have a great military. They were awesome traders. One cannot sustain an army on a bad economy. The scale of the Roman economy would mean that the later Roman imperial slave trade would dwarf anything else humanity would see in terms of slavery in coming ages.

So, to recap;

• When Roman armies captured slaves what did they do? Gave them to enlisted men (enlisted men used to lighten the text, just think "anyone in the army"), who then sold them on to traders or to other Romans directly or to anyone who wanted a slave really... "Gave them" is also kinda misleading since the new "owners" may have captured the slaves themselves so there isn't a need to "give" the slaves to them..

• Did they sell them directly to other Citizens or some sort of slave company? Both!

• How big was this as a source of income for the Roman state? It had very light barriers to entry, making it a dynamic trade, but the sheer scale of the operation means it was one of the cornerstones of the Roman economy and thus important to the state.

Sources:

SCHEIDEL, W. (2004b) ‘Human mobility in Roman Italy, I: The free population’, JRS 94: 1-26 SCHEIDEL, W. (2005a) ‘Human mobility in Roman Italy, II: The slave population’, JRS 95 (in press) http://www.princeton.edu/~pswpc/pdfs/scheidel/050704.pdf