"The Lord's Prayer" (Matthew 6:9-13; Luke 11:1-4) seems to be inherently public, communal (Michael Wilkins, The NIV Application Commentary: Matthew), liturgical (R.T. France, Tyndale New Testament Commentaries: Matthew) and intimate (family oriented) in nature.
Jesus taught a variation of this prayer to his disciples in answer to their request to teach them to pray (maybe the first time they heard him speak of it?), after having witnessed his own praying to his Father. So this prayer can certainly be prayed in private. But it is addressed to disciples (I include us today) in plural pronouns and the prayer itself is about "us", not just "me". That together, with the fact that this prayer seems patterned after the Jewish liturgical prayer called the Kaddish (Scot McKnight: The Jesus Creed: Loving God, Loving Others) seems to me to indicate that this prayer is to be done in public together as "the community of Jesus".
This prayer goes back to "the Jesus Creed". We're to love God with our all, and to love our neighbor as ourselves. And the prayer Jesus taught falls out like that. Again, like Jesus' changing of Shema to add love of others to love of God (Mark 12:29-31) (Scot McKnight) we see that this prayer consists of petitions concerning love to God and love to our brothers and sisters in the community of Jesus.
Two thoughts I'd like for us to consider here. Who are we praying to? And what does that mean about us?
Jesus taught us to pray: "Our Father"/"Father". Jesus certainly used other titles when addressing God or referring to God, in keeping with the Jewish practice of his day (example: Matthew 11:25 though even here he includes "Father"; Matthew 26:64; 27:46). But overwhelmingly he used pater, and on occasion abba both translated into our language "Father", when praying or referring to God. This emphasis is unique to Jesus up to that time in Scripture. And when we pray, we're to refer to God as "Father".
Yes God is holy, to be held in awe, a consuming fire (Hebrews 12:28,29). But as Jesus taught us we're to look at God and approach him as Father. Jesus as our Brother (Hebrews 2:11-12). Certainly this does not not exclude Jesus as Lord which is always true, but not our focus here. The Spirit is our Advocate, or Helper in all of this (John 14-16).
Maybe God does seem distant to us many times, or especially at certain times and points in our lives. But in Jesus we are to address God as Father. This means we are his children. And as his children we are family together. For better or for worse. God wouldn't let it be any other way.
Do we see God as "our Father" and Jesus as "our Brother"? Do we see each other as brothers and sisters in Jesus, as God's very own family? This is at the heart of this prayer that we are to pray. And the prayer is surely meant to be for us a powerful reminder and enactment so as to help establish this among us.