I keep thinking that in order to let children know that they can pray in their own words, the parents would have to demonstrate it... even though I never heard my parents pray in their own words. They taught me the prayer to my guardian angel, and all the prayers in the rosary, and that's what we prayed at home.
I also knew that when I prayed before Mass, it could be any prayer I knew, or any prayer I could come up with. If I was told to pray for someone, I did so silently, in my own words. I guess these things were enough to show me that private, extemporaneous prayer was all right? However, learning this way didn't teach me to, for example, say "Thy will be done" after a request. My personal prayers are also typically a big mess of thoughts without much coherence. I never learned to organize them, and I'd have great difficulty saying them out loud.
This is why I'm still curious to hear how other families pray in the home.