I am a member of the Church of Christ in a local congregation. Of late, when we pray in the church, we do allow individual members some time to say their prayers adding that we should not speak loud. I am perplexed now.
I don't blame you for being perplexed. Such prayers would be individual efforts done at the same time and not joint efforts by the church. While there is nothing wrong with individuals praying, praying can be done at any time and in any place by Christians, and such should be emphasized since that is what God teaches. The church coming together for worship is for sharing worship together. "For we, though many, are one bread and one body; for we all partake of that one bread" (I Corinthians 10:17). It is also for building each member up. "Let all things be done for edification" (I Corinthians 14:26). I know of no command or example of the early church praying together silently and individually.