The Senators and Representatives, Not the President of the Senate

The Constitution says senators and representatives, not the president of the Senate. They shall in all cases except treason felony and breach of the peace be privileged from arrest during their attendance up session of their respective houses. And for any speech or debate in either house, they shall not be questioned and any other place. The question is who's privilege is it? Is it the president, meaning it follows, you know, to a former president?"