The Catholic Church permits the Orthodox to receive the Eucharist in a Catholic service, but cautions them to observe their own disciplines. For example, OSCCB offers:
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).
The Catholic Church permits a Catholic to receive in an Orthodox service if a Catholic service is not available:
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.(canon 844 §2)
The appropriate sections of the canon law can be found here.
According to the Orthodox Church, the Orthodox may not receive the Eucharist in a Roman Catholic Church. For example, the OCA offers this answer:
Further, Orthodox Christianity does not permit its faithful to receive Holy Communion in non-Orthodox communities, whether they be Roman Catholic, Protestant, or whatever. Hence, while Roman Catholicism may extend Eucharistic hospitality to Orthodox Christians, it does not mean that Orthodox Christians are permitted to accept such hospitality.
Further, non-Orthodox can not receive in an Orthodox service:
Because a non-Orthodox individual has chosen not to embrace all that Orthodox Christianity holds, the non-Orthodox individual makes it impossible for an Orthodox priest to offer him or her communion.
Before the Great Schism, there was no formal separation between East and West.