In music of the Baroque period, trills do typically commence on the upper auxiliary (so what you describe sounds correct).
When voices played by the left hand and right hand happen to coincide on the same note, you're generally not expected to play it with both hands. Take into account the duration and articulation of the "shared" note, and use the hand that makes most sense in context (i.e., most efficient execution and least disruption to the flow of the music).