The Roman Empire maintained a free grain dole for much of its existence. How was this funded? Was there any stigma against people who took advantage of this form of public assistance?

Ah, the "grain dole." Generally misunderstood yet frequently talked about--for the realities of the so-called "dole" see here (I have a more recent, better thread on the same topic, but it looks like I managed to misplace it somehow :/)

Anyway, the distribution of public or state-subsidized grain had several stages. There was only one period in which all (adult male) citizens were eligible to receive their 5 modii per month (not really very much) for free, following the Publius Clodius' lex de frumentaria. The major protests against the bill, such as they existed, were that it would drain the public treasury--Clodius, however, packaged the bill with the annexation of Cyprus and seizure of the king's wealth, claiming that it would offset the losses to the treasury. Whether it did or not is unclear. What is known is that though the lex Clodia apparently got off to a good start it was plagued very quickly by shortage. What exactly caused the shortage we're not too sure on. Tatum rejects the idea that Clodius' bill actually did drain the treasury and ascribes the shortage to poor harvests--poor harvests do not entirely convince me. I'm more inclined to suppose that speculation among the grain traders was the culprit, as was so often the case in previous famines--grain traders would trade in futures or stock up enormous piles of grain that they wouldn't sell to create artificial scarcity so that they could jack their prices into the roof, and the lex Clodia failed to impose legal limits on the activities of the grain traders. In 57 Pompey was given powers of curator annonae, along with a grant of HS 40 million to purchase grain, along with the right to oversee distribution for the next five years. How exactly Pompey managed this, on the lowest levels of distribution, we're not told, though there was a decent amount of turmoil over a possible expedition to Egypt to restore Ptolemy XII, with the expectation of additional grain supplies (the expedition never occurred--Ptolemy was restored by the proconsul of Syria, Gabinius, who was later acquitted for having illegally exited his province). Caesar lowered the recipients of public grain to 150,000, who were apparently kept on a list, with a waiting-list of sorts put together as well, from which new recipients could be drawn when the old ones died (which, given the shockingly high mortality rate among the urban poor, was not necessarily a long wait).

The major reform to the public grain distributions occurred under Augustus, who established the cura annonae as a permanent duty. Under Augustus, who held ultimate power over the cura, the system underwent several reforms--initially it was overseen by two praetorians, then two consulars, and by Augustus' death the praefectus annonae was an eques, in which order the post would remain throughout the Principate. Augustus introduced tickets, tesserae, that were held by the 200,000 who were included on his lists, and could be inherited or sold. I should stress that the lists of recipients was not at all directed at the "needy." We're not totally sure what the requirements were but the public grain was limited to inhabitants of the city, excluded women and children, and probably only included Italians and freeborn persons--this last cuts out a hugely significant demographic group, since the proportion of freedmen within the urban plebs was absolutely enormous. The very fact that tickets could be bought and inherited (but were otherwise inalienable) indicates that within a couple generations the recipients of public grain more or less made up their own little unit within the urban plebs.

These tickets undoubtedly helped the issue of distribution and supply greatly. The grain that was provided at the public expense was both taken from tax and purchased on the market by the state. Most provinces paid some sort of grain tax to the city--Sicily's, Sardinia's, and Africa's (and later Egypt's) being the most significant. These taxes alone were generally not enough to supply the city apparently, since the curatores annonae would still have to purchase from the public treasury. The private grain trade was one of the largest and most productive industries in the Roman world. Hundreds of thousands of ships carried millions of modii plied the Mediterranean, and with them came private merchants who traded in this vital commodity. Under the Republic the price of grain was notoriously flexible, because of the freedom of the grain traders to set their own prices and their manipulation that I already mentioned--even in times of actual shortage, like when the Cilicians were threatening the grain routes from Sicily and Africa, grain traders would further reduce the supply to a mere trickle, thus ensuring that in the future they would be able to maintain their supply and keeping their prices even higher than the famine would require. Under Augustus this came to an end with the passage of the lex Julia de annona, which imposed penalties on grain speculation and stockpiling--later Claudius also introduced a measure to reimburse grain haulers whose ships had been lost, thus providing an incentive to carry grain. This, more than anything else, probably streamlined the public grain supply. It ensured that grain purchased by the state in addition to taxed grain would be purchased at a relatively stable rate (barring catastrophic crop failure) and it also provided that the additional purchase of grain on the market would be within reasonable limits. And make no mistake, the urban poor had to purchase grain on the market. The public grain distributions were limited to 5 modii per month, which was probably enough for an individual to get by, but not nearly enough for a family--and it only included adult males, probably heads of household, so multiple individuals in the same family could not be provided with separate portions. In fact part of the problem with the famine in 57 was probably that when the urban poor, not only the state, was unable to get their additional grain on the market, for whatever reason the prices were raised.

Attitudes towards the public grain sort of vary. In the Principate we find people like Juvenal railing against the degeneracy of the urban population, but Juvenal's kind of an asshole and he thinks everyone (especially Greeks and women) are degenerate. Generally speaking the attitudes of our authors towards the urban plebs in general are not kind, and into this group we may include recipients of the public grain--like the rest of the urban plebs they would've been lumped in with the general "idler" population that people like Tacitus disliked so much (nevermind that the "idle" nature of the Roman urban population is a complete fabrication). Still, when given the opportunity, the wealthy and powerful had shown themselves in the past perfectly happy to receive publicly subsidized grain. During the passage of the lex Clodia de frumentaria wealthy masters of households throughout the city began freeing their slaves in huge numbers, so that the new freedmen would receive their portions of the public grain and provide some of it to their former masters as clients.