This section is from the book "The National Capitol. Its Architecture Art And History", by George C. Hazelton, Jr. Also available from Amazon: The National Capitol Its Architecture Art and History.
At the organization of the two Houses of Congress, a question arose as to the proper method by which bills and communications should be transmitted from one to the other. The matter was referred to a committee; and it was agreed that in the interim such communications should be conveyed by the Secretary of the Senate and Clerk of the House. The report of the committee was not adopted; and the practice, which began as a temporary arrangement, has become customary. It has been disregarded in two instances. In 1813 the Embargo Act was sent to the Senate by two of the Members of the House, with a request that the Senate consider it confidentially; and the bill was reported by the Senate to the House in like manner. The second instance was in 1815.
Communications from the President to Congress were at first delivered by Cabinet officers, but the President's private secretary early became the messenger; and one of his secretaries still continues to perform this important duty. Communications from the Senate to the President are made through a committee of Senators or by its Secretary; from the House by a committee of Members or by its Clerk.