In the Western Church, there is a distinction between “public” and “private” prayer. In the West, the term “Liturgy” is synonymous with “public” prayer, the “prayer of the Church”. Liturgy, in the West, consists of the Mass, the Sacraments, the Divine Office, formal blessings (including exorcism) and certain approved devotions (e.g., the Litany of the Saints) which may be used during those services. All of these services *ought* to have a priest officiating, though deacons can officiate certain sacraments and blessings, and laity may officiate the Divine Office. When one participates in “public” prayer, one prays with the Church. If a priest says a “private” Mass by himself or with a few others, he is still engaging in “public” prayer. If a woman prays the Divine Office alone in her room, she is still engaging in “public” prayer.
By contrast, any other form of prayer is “private.” For example, the Rosary is a private devotion. It is the highest of private devotions, but it is still a devotion. If 1000 people are praying the Rosary together, it is still, in the technical sense, “private.” Then again, there are “private” devotions approved for “public” use (such as the Rosary), but that’s a technicality beyond our purposes here.
In any case, I don’t know if there’s a Western equivalent of the Moleben, and I don’t know how such an equivalent would be understood. The East does not understand these terms—public, private and liturgy—the same way. Only the Eucharist is Liturgy in the East. The services considered “liturgies” in the West—the Sacraments, the Office and formal blessings—are required to have priests officiating in their complete form. All of them have parts that are to be omitted if there is no priest (even a deacon cannot say those parts). The Moleben counts as such a service, and would probably be considered a “liturgical” or “public” devotion in Roman terminology.
In any case, because there’s no equivalent, and because it provides a devotional link to our Eastern brethren, I strongly recommend practicing this powerful devotion. It is based upon the model of the Divine Office. In a full Vigil, one says Molebens and other prayers between Vespers, Compline, and Matins. The moleben is said in honor of Jesus, Mary or a particular saint, usually in front of the Icon of the one being honored. It also involves places to pray for a specific intention, and the closing prayer includes a reference to the saint of the day. So, even if one is saying, say, the Moleben to Jesus, it provides opportunity to honor the saint of the day, particularly on Sunday, when the saints otherwise get superseded.