They didn't. What makes you think they would?

Jesus was born on the margins of the empire, in a subjugated territory, to a relatively poor, rural family, and his birth made almost zero impact on contemporary affairs. There is no record, nor reason to suspect, that Roman leaders, senators, or Caesar Augustus would have heard of, or cared about, such a birth.

The only reason we have anything resembling documentary evidence of Jesus' birth is the (still largely localised) impact of Jesus' adult ministry and the events surrounding his death, which prompted the somewhat later writings of the Gospel accounts which do show an interest in his birth, and may record oral recollections of persons who remembered events surrounding his birth.