I think the primary issue when posing this question is reframing our thinking in regards to how we perceive citizenship. While currently we define citizenship primarily through proof that one was born in a certain country, Roman citizenship was quite different. Important to remember in this context is that while there was mobility in the Roman Empire, this was by far the exception rather than the rule. Most Romans were deeply connected through social ties, through a family, or a gens. If a person needed to prove citizenship they needed to do no more than ask their community to produce a witness, and this would be the most common form of "proof".

However, during the turn of the millenium (under the rule of Augustus), a birth database was created. The database was not mandatory, but if one was registered, they would receive a wooden diptych was seven inches high and six inches wide. Written on the waxed surfaces of this document were the date of birth, the name of seven witnesses and the abbreviation q. p. f. c. r. e. ad k. (the letters c.r.e meant cieum romanam/num exscripsi/t) indicating the possession of Roman citizenship. The diptych could be used for life to prove citizenship and was written only in Latin until the time of Emperor Severus (222-235 AD) . We have proof of this thanks to several archaeological finds discussed in (1), which also contains far more detailed information about the database.

The more interesting case presents itself when we look at the "edge cases", i.e. the people on the frontiers (the so called barbarians) that began to populate the empire in the later parts of the 1st millenium AD. What is clear is that Roman citizenship became significantly less hard to obtain as the empire grew ever larger. Mathisen summarizes it best with quotes from a few of the most prominent primary sources.

As the centuries wore on, the Roman citizen body continued to expand. Roman citizenship, and access to Roman ius civile, became less a special status and more a lowest common denominator. This process culminated in the issuance in 212 C.E. of the Antonine Constitution, in which the emperor Caracalla (211–217) granted citizenship to nearly all of the remaining free peregrini in the Roman Empire. The only surviving copy of Caracalla’s law, a papyrus Greek translation of the Latin original, is very fragmentary, but the crucial words are clear: “I grant to all those in the Roman world the citizenship of the Romans.” All that can be said with any certainty about the rest of the text is that people known as dediticii were excluded from citizenship.

Due to the expanding base of citizens, people especially at the edges of the empire began to identify less with the broader Roman Empire and more with their province (referred to as "provinciales"). These changes in the Empire also meant that there were less benefits to a citizen and more benefits from social class alone. Just as Roman citizenship became easier to obtain, it became harder to reap the same benefits as early in the Empire. Especially after the Caracalla's decree we find that the benefits associated with citizenship began to deteriorate.

With regard to specifically barbarians or outsiders obtaining citizenship, we have only a few sources, which again Mathisen expertly compiles these disparate sources,

In 383, the orator Themistius opined that Goths were “no longer called barbarians but Romans,” a sentiment seconded by Pacatus, who stated in 389 that Theodosius I (379–395) had ordered defeated barbarian soldiers to “become Roman.” It is difficult to see how “becoming Roman” would not entail having some degree of Roman citizenship. Citizenship specifically is mentioned by Claudian, who, in his panegyric to Stilicho in 400, observed of Rome, “She calls together as citizens those whom she has conquered.” This view survived in the mid-sixth century, when Corippus could say, in a North African context, “Whatever gentes the Roman Empire sees being faithful and subject, it considers them to be Latin citizens.” More specifically, the rhetor Synesius stated that Theodosius I “considered the Goths worthy of citizenship.” The only person of barbarian ancestry specifically called a Roman citizen is the Master of Soldiers Stilicho, the son of a Roman mother and a Vandal father, of whom Claudian said, “Rome rejoiced that she deserved to have you as a citizen.”But how Stilicho gained this status—whether through his mother, through some action of his own, or by default—we are not told. We still, therefore, have no examples of barbarians actually being “made” citizens; nor do we know what kind of reality lies behind the rhetoric.

"Barbarians" were subject to some Roman law, occasionally being forcefully resettled and were taxed. However, they also received benefits. Barbarian soldiers received veterans benefits, and the ability to marry into Roman families. However, much of this discussion extends far beyond the period of "bread and circuses" which your question was addressing, so I have left some sources below where you can find further information on the exact contours of the legislature regarding barbarians as the Empire expanded.

TL;DR: Early Romans relied on primarily familial and community connections to verify citizenship although there was a registry. As the Empire expanded, the definition of citizenship grew looser and the benefits associated were distributed based on social class instead.

(1)Schulz, Fritz. “Roman Registers of Births and Birth Certificates.” The Journal of Roman Studies, vol. 32, 1942, pp. 78–91. JSTOR, www.jstor.org/stable/296462. Accessed 7 July 2020.

(2) Mathisen, Ralph W. “Peregrini, Barbari, and Cives Romani: Concepts of Citizenship and the Legal Identity of Barbarians in the Later Roman Empire.” The American Historical Review, vol. 111, no. 4, 2006, pp. 1011–1040. JSTOR, www.jstor.org/stable/10.1086/ahr.111.4.1011. Accessed 7 July 2020.

Addtl. Resources:

Cherry, David. “The Minician Law: Marriage and the Roman Citizenship.” Phoenix, vol. 44, no. 3, 1990, pp. 244–266. JSTOR, www.jstor.org/stable/1088935. Accessed 7 July 2020.

Steel, Catherine. “Roman Citizenship between Law and Practice.” Enfranchising Ireland?: Identity, Citizenship and State, edited by Steven G. Ellis, Royal Irish Academy, Dublin, 2018, pp. 7–18. JSTOR, www.jstor.org/stable/j.ctvgd1dd.4. Accessed 7 July 2020.

/u/mayaxs is entirely right that the question of citizenship is a bit of a thorny one right now. I should say, however, that this question is actually pretty simple.

In point of fact in the Principate the five modii of state-subsidized grain (about enough for a person and a little more) per month per head of household was not available to all citizens. The lex Clodia in 58 BC had granted 5 modii per month of state-subsidized grain for all citizen heads of household, but this was short-lived. Pompey as curator annonae reduced the number of recipients to 320,000, presumably drawn up in a register, and Caesar reduced that to 150,000, maintained on a list kept up to date by the praetors. Under Augustus and subsequent regimes the number of recipients fluctuated, though it never reached the height it had under Pompey. The state-subsidized grain was not the right of all citizens, but the privilege of a relatively few, known by their possession of lead tickets, or tesserae. These, by the first few decades of the Principare, were inalienable, and could be passed along in inheritance or sold.

I've never heard of games being limited only to citizens, but entry to certain games was also indicated by the possession of similar tesserae

I appreciate your response very much. Thank you.