So what does Rome teach about non-Roman Christians communing there?

From John Paul II:


"While it is never legitimate to concelebrate in the absence of full communion, the same is not true with respect to the administration of the Eucharist under special circumstances, to individual persons belonging to Churches or Ecclesial Communities not in full communion with the Catholic Church. In this case, in fact, the intention is to meet a grave spiritual need for the eternal salvation of an individual believer, not to bring about an intercommunion which remains impossible until the visible bonds of ecclesial communion are fully re-established."
 and

"In this context, it is a source of joy to note that Catholic ministers are able, in certain particular cases, to administer the Sacraments of the Eucharist, Penance and Anointing of the Sick to Christians who are not in full communion with the Catholic Church but who greatly desire to receive these sacraments, freely request them and manifest the faith which the Catholic Church professes with regard to these sacraments. Conversely, in specific cases and in particular circumstances, Catholics too can request these same sacraments from ministers of Churches in which these sacraments are valid. The conditions for such reciprocal reception have been laid down in specific norms; for the sake of furthering ecumenism these norms must be respected."

from Canon Law:

"If the danger of death is present or if, in the judgement 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."

From Benedict XVI:

"56. The subject of participation in the Eucharist inevitably raises the question of Christians belonging to Churches or Ecclesial Communities not in full communion with the Catholic Church. In this regard, it must be said that the intrinsic link between the Eucharist and the Church's unity inspires us to long for the day when we will be able to celebrate the Holy Eucharist together with all believers in Christ, and in this way to express visibly the fullness of unity that Christ willed for his disciples (cf. Jn 17:21). On the other hand, the respect we owe to the sacrament of Christ's Body and Blood prevents us from making it a mere "means" to be used indiscriminately in order to attain that unity. (172) The Eucharist in fact not only manifests our personal communion with Jesus Christ, but also implies full communio with the Church. This is the reason why, sadly albeit not without hope, we ask Christians who are not Catholic to understand and respect our conviction, which is grounded in the Bible and Tradition. We hold that eucharistic communion and ecclesial communion are so linked as to make it generally (my emphasis) impossible for non-Catholic Christians to receive the former without enjoying the latter. There would be even less sense in actually concelebrating with ministers of Churches or ecclesial communities not in full communion with the Catholic Church. Yet it remains true that, for the sake of their eternal salvation, individual non-Catholic Christians can be admitted to the Eucharist, the sacrament of Reconciliation and the Anointing of the Sick. But this is possible only in specific, exceptional situations and requires that certain precisely defined conditions be met (173). These are clearly indicated in the Catechism of the Catholic Church (174) and in its Compendium (175). Everyone is obliged to observe these norms faithfully."

My own comments:

Because there has been confusion over what the Roman Catholic Church says or does not say about those outside the umbrella of Rome receiving the Eucharist (or other sacraments) from a Roman Catholic priest or parish, I am indebted to a former Lutheran, Australian David Schutz, now Roman Catholic, who has provided the appropriate and authoritative words on the subject.

I might note that the words of John Paul II in the first two citations sound pretty much like the Lutheran Church Missouri Synod in its stance toward intercommunion between churches in general and its practice of close(d) communion with pastoral discretion in particular.  It would seem that in this there is great affinity between the positions of the Roman Catholic Church and the LCMS in this area and that those who practice open communion or who view Eucharistic sharing as a means to greater unity of doctrine and confession stand on the outside of mainstream catholic and universal practice both in antiquity and now.