If leaders of Roman Catholic and Orthodox churches are still far from resolving the theological issues separating these two true churches, is there any ecumenical movement at lay people level like the Taize fraternity where Protestants and Catholics live and worship together? Are there Christians from both sides who recognize the truthfulness of the other side and are longing to share the richness of these two traditions despite the opinions of their respective church hierarchies?
One such organization is Orientale Lumen.
As stated on their web site here:
Started in 1997 in Washington, DC, these ecumenical conferences are a "grass roots" movement among lay persons and clergy to provide a forum for Christians to learn about the "light from the east." They allow Eastern Orthodox, Eastern Catholics and Roman Catholics to meet and pray together, learn from each other's traditions, and become friends together searching for a common goal: "that they all may be one" in the One Church of Christ.
The conference speakers usually include members of the clergy, so I wouldn't call it a "ecumenical movement at lay people level", but it is an excellent conference, which addresses topics such as you suggest.