Only Roman Catholics (e.g. Catholics in communion with the Pope) can receive. Orthodox (e.g. Catholics not in communion with the Pope) are allowed to receive according to canon law of the Roman Church, but are not allowed by their own (Orthodox) authorities. More generally, Christians who believe what the Roman Church teaches regarding the Eucharist (this can be strictly interpreted as “transubstantiation” [excluding everyone] or loosely as “real presence” [including Orthodox, Lutherans, and High Church/Anglo-Catholics]) and are unable to access a Church of their own denomination for a “significant period of time” (again open to interpretation) may be allowed to communicate under “extraordinary circumstances” (cf Canon 844 §§1-4), which I believe is defined in the Catechism, ¶1401:“When, in the Ordinary’s * judgment, a grave necessity arises, Catholic ministers may give the sacraments of Eucharist, Penance, and Anointing of the Sick to other Christians not in full communion with the Catholic Church, who ask for them of their own will, provided they give evidence of holding the [Roman] Catholic faith regarding these sacraments and possess the required dispositions.”
So, strictly interpreted, the Roman Church allows only Roman Catholics and Eastern Orthodox Catholics to receive communion, but the Orthodox are excluded by their own authorities (an Orthodox is not allowed to receive Roman communion). Less-strictly interpreted, high-church Anglicans and Lutherans can receive in certain situations. As commonly (mis)interpreted in several parishes I have seen, anyone can receive at any time as long as they profess to be Christian, or at least of a moderately ritualist variety (cf above poster, “Maybe it’s just up to the local pastor (in practice)?” indeed, it is so in practice, as are other liturgical abuses: the ordinaries very rarely legislate liturgical abuse). It appears “acceptable practice” falls somewhere between the strictest interpretation and the misinterpretation.
From the 1996 NCCB (now USCCB) guidelines: “… Members of the Orthodox Churches, the Assyrian Church of the East, and the Polish National Catholic Church are urged to respect the discipline of their own Churches. According to Roman Catholic discipline, the Code of Canon Law does not object to the reception of communion by Christians of these Churches (canon 844 § 3) …”
[quote=Codex Iuris Canonici 1983]Canon 844 §1. Catholic ministers administer the sacraments licitly to Catholic members of the Christian faithful alone, who likewise receive them licitly from Catholic ministers alone, without prejudice to the prescripts of §§2, 3, and 4 of this canon, and -> can. 861, §2.
§2. Whenever necessity requires it or true spiritual advantage suggests it, and provided that danger of error or of indifferentism is avoided, the Christian faithful for whom it is physically or morally impossible to approach a Catholic minister are permitted to receive the sacraments of penance, Eucharist, and anointing of the sick from non-Catholic ministers in whose Churches these sacraments are valid.
§3. Catholic ministers administer the sacraments of penance, Eucharist, and anointing of the sick licitly to members of Eastern Churches which do not have full communion with the Catholic Church if they seek such on their own accord and are properly disposed. This is also valid for members of other Churches which in the judgment of the Apostolic See are in the same condition in regard to the sacraments as these Eastern Churches.
§4. If the danger of death is present or if, in the judgment of the diocesan bishop or conference of bishops, some other grave necessity urges it, Catholic ministers administer these same sacraments licitly also to other Christians not having full communion with the Catholic Church, who cannot approach a minister of their own community and who seek such on their own accord, provided that they manifest Catholic faith in respect to these sacraments and are properly disposed.