I've written on the so-called "grain dole" before, and you might check it out. A note: there was no "bread dole." The Roman state subsidized the purchase of grain, which was distributed at a rate of 5 modii per month to (almost certainly) qualifying heads of household. The state did not distribute bread, nor was the state-subsidized grain milled. This is, broadly speaking, in line with general practice in Roman Italy and most of the ancient Mediterranean, where grain was typically purchased in its unmilled state at market, and was always stored unmilled.

We have no evidence of any fraud of the type that you're describing, namely the claiming of state-subsidized grain by those unqualified or who had already collected their 5 modii for the month. No ancient source gives any indication that this happened. We might hypothesize why, since surely such fraud did occur. The most obvious explanation, that the sources simply aren't interested in the behavior of the ordinary people, doesn't hold up precisely because the sources are in fact extremely interested in the state-subsidized grain from the moment of its institution. We're told of one type of fraud--the mass manumission of slaves by wealthy slaveholders prior to Pompey's cura annonae--, so defrauding the state of its grain was a known problem. Considering the great controversy surrounding the passage of the lex Clodia, chief among which was that the state couldn't afford it, it's rather remarkable that "double dipping" by the common man isn't mentioned anywhere ever as a justification for regulating the state distributions. Instead, it's the wealthy who our sources point to as abusing the system.

The trouble is that we know effectively nothing about how the grain was distributed. This is probably due less to disinterest on the part of the sources but rather that the information was obvious. Much of our knowledge of the state-subsidized grain comes from sources like Cicero's public speeches. In such contexts it would be unnecessary to explain how the grain was distributed, since the audience already knew. We know that Pompey drew up some kind of list based on some sort of qualifications that restricted the number of people eligible to merely most of the urban citizen population rather than all of it. Under Caesar the praetors were supposed to maintain a list of people eligible, and it looks like the lex Clodia also placed the urban praetor in charge of distribution, although it's not clear whether he also would've been placed in charge of maintaining a register following Pompey's cura annonae. Under the emperors a more robust system of overseeing the register and proving one's eligibility by use of metal tickets developed as the state-subsidized grain became increasingly restrictive. Similarly, the imperial officers in charge of the grain supply in general (including the state-subsidized grain) became a highly developed administrative position.

The grain must have been distributed at multiple locations, but we don't know how. All attempts to reason out who actually distributed grain on the ground are unconvincing. Nor do we really know how often, at least in the Republic, state-subsidized grain was distributed. It seems likely that all 5 modii were distributed at once, which suggests that grain was distributed at a set date or series of dates, but there are impediments to that suggestion as well as a complete and total lack of direct evidence of any kind. It seems most likely that in the early days of the lex Clodia the state-subsidized grain was probably distributed by local organizations, which seems like the only way that long-term residents could be identified. Alternatively, tribal organizations may have been used. A system existed for the distribution of campaign funds to the tribes from electoral hopefuls from their own tribe, and such organizations might potentially have been able to provide the manpower necessary. But there's exactly zero evidence for any of this, and our knowledge of local organizations during the Republic is extremely spotty at best. The suggestions I've just given also don't seem to fit very well with what little we know about local organizations in the Republic, which were typically suppressed rather than encouraged, and fit much better with the more centralized local administration of the empire, which was able to control local organizations and associations more systematically than had been possible under the Republic by tying them directly to the imperial administration. What little we can say is basically that the logistics of the state-subsidized grain in the imperial period was clearly much more centralized and sophisticated than in the Republic. Yet at the same time, fraud and illegitimate abuse of the system don't seem to have been terribly common even in the years immediately following the lex Clodia.