I was raised in the Catholic Church (...insert "Darth Vader's Theme" from "The Empire Strikes Back," here...no, I'm not serious...) and there the term "ecumenism" was considered to be properly defined as referring to the action of the Holy Spirit upon Christians to seek full, visible unity in teaching and spiritual life. So, from this point of view, "ecumenism" is a Christian "in-house" affair.
That does not mean that Roman Catholic-raised people do not seek constructive relationships with non-Christians. There is simply a different term used for it: "interreligious dialogue." (That said, the late Pope John Paul II once noted that the term "ecumenism" is being used for what was considered "properly" to be interreligious dialogue...and he didn't exactly seem to mind.)
As for myself, where interreligious dialogue is concerned, I generally follow my own understanding of the Christian interreligious theology of the late Fr. Jacques Dupuis. In a nutshell, God is not ashamed to work with people who are in other religions, and even through those religions. The "God" Who so works is expressly the Christian God--the Father, the Son and the Holy Spirit. That said, the "other religions" are looked upon as "gifts of God to the nations." While Fr. Dupuis (...and Pope John Paul II...) would say that those other religions contain errors, gaps, and insufficiencies (...remember, this is from a Roman Catholic viewpoint...) my view tends to be that *all* religions have gaps, insufficiencies, and errors. Thus, I would say, that inspiration and prophecy *could* come from any source...as can error. The discernment of prayer and conscience is necessary in any event, to be able to separate the gold from the dross, so to speak.
As for conservative/orthodox Christians...try to be patient with them. The answers they are seeking (...when they are at their best...) are ones that do not involve the surrendering of religious principles. That can be tough, especially when the obvious answers seem to violate those principles. Fr. Dupuis himself was under investigation for a while by the Sacred Congregation for the Doctrine of the Faith, as headed by then-Joseph Cardinal Ratzinger, now Pope Benedict XVI. That investigation was initially something of a heresy-hunt...and at the end of it all, the only thing that was actually done was to say that FR. Dupuis seemed to be unclear about certain ideas...which he most expressly affirmed, and included, in his subsequent theology efforts. To my knowledge, Fr. Dupuis was never bothered again. So, again, I would counsel patience. I won't sit here and tell you that it's easy or quick, but sometimes you do live to see some good results.