The Senators and Representatives shall receive a compensation for their Services, to be ascertained by Law, and paid out of the Treasury of the United States. They shall in all Cases, except Treason, Felony and Breach of the Peace, be privileged from Arrest during their Attendance at the Session of their respective Houses, and in going to and returning from the same; and for any Speech or Debate in either House, they shall not be questioned in any other Place.
Article I Section 9
The privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it.
Article IV section 2.1:
The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.
Amendment XIV:
Section 1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State [...]
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I personally don't care for textualist arguments, but the concept of privilege is explicitly mentioned 4 times in the Constitution.
EDIT: I'm getting downvoted for pointing out a fact, with citations. Impressive.