Catholic theology states that obviously the body and blood of Chirst are equally present in both the bread, once it has been transformed into the Body of Christ and in the wine, once it has been transformed into His Blood.
There is therefore no neccessity for communicants to receive both. Since Vatican 2 they can do so, if they so desire, although this usually only happens at the larger church festivals, or conversely when Mass is being celebrated for a small congregation, due to the difficulty of keeping a large consecrated but unconsumed amount of wine.
For centuries prior to Vatican 2, only the celebrant - the priest who actually consecrated the bread and wine, partook of the wine, although both bridegroom and bride do so in the course of their nuptial Mass - but this is because they are actually the celebrants of the sacrament of marriage, not the priest - he is witnessing their vows, and consecrating the Eucharist.
Popular Catholic tradition holds that the wine having transformed into the Blood of Christ guards against infection, but this cannot be proved scientifically. It hardly matters, as even at a Mass where both bread and wine are offered to the congregation, you decide yourself whether you wish to partake of both, or not.
From the doctrinal point of view it makes absolutely no difference whether the wine is consumed from the same chalice or not. If it is, the chalice is wiped after each person has sipped from it - which again may well not be any very effective safeguard.
Traditionally it was thought that Christ and his disciples drank from the same cup at the Last Supper. Whether they did so or not, no-one can possibly say.
No modern Jewish family celebrating the Passover would all drink from a shared cup, and it was the Passover Jesus was celebrating, although he gave the ritual a new (and to most Jews fallacious) significance.
The Anglican and Lutheran churches will usually accept it if for health reasons you prefer only to receive either the Bread or the Wine.