Whose Prayers Does God Hear?

So much of the language of Christianity and other religions is exclusive with rewards - blessings and salvation - either exclusively or predominantly for members and insiders. Perhaps chief among those rewards is the privilege (and promise) that the Sovereign God hears our prayers. And that even if God hears the prayers of all humanity, we who are in special (or even right) relationship with God have special, intimate access. Many earnestly invite others into that special relationship, but there is still an us/them, insider/outsider dichotomy.
There is a passage in the Jewish and Christian scriptures that models a radically different practice. In his much celebrated prayer marking the Divine Habitation of the temple when God physically moves into the temple in Jerusalem in the Ark of the Covenant, Solomon entreats God to hear the prayers of foreigners.
1Kings 8:41 “When a foreigner, who is not of your people Israel, comes from a distant land because of your name 42 —for they shall hear of your great name, your mighty hand, and your outstretched arm—when a foreigner comes and prays toward this house, 43 then hear in heaven your dwelling place, and do according to all that the foreigner calls to you, so that all the peoples of the earth may know your name and fear you, as do your people Israel, and so that they may know that your name has been invoked on this house that I have built..."
What strikes me is that Solomon isn't praying for the conversion of the outsiders. He does pray that foreigners would know and reverence (fear) his God. But not that they would adopt the religious practices and beliefs of his people. It seems that many religious people and communities practice the opposite: once people have joined or converted, then they are assured that God will hear them.
What would this world be like if we all prayed that God would hear and answer the prayers of people who don't believe what we believe or worship how we worship?
There is something else in this text, Solomon imagines a world in which outsiders find prayer to his God desirable. Do the pious folk in any religious tradition - particularly those that advocate conversion - conduct themselves (ourselves) in such a way that anyone outside our communities actually wants to pray to our God?