In plain English
FAR 52.224-1, Privacy Act Notification, is a contract clause prescribed at FAR 24.104, most recently dated Apr 1984. The complete official text is reproduced below, verbatim, from GSA's published FAR source files.
Its text contains no sentence requiring insertion into subcontracts; see the flowdown section below for what that does and does not mean.
Does it flow down to subcontracts?
No flowdown mandate found in the clause text
We scanned the full clause text and found no sentence directing the contractor to insert this clause into subcontracts. That is a statement about the text, not legal advice: a prime contractor may still flow terms down contractually, an agency supplement (DFARS, VAAR, …) may add requirements, and clauses listed below (if any) may order this clause into subcontracts from the outside.
Where it's prescribed
As prescribed in 24.104 , insert the following clause in solicitations and contracts, when the design, development, or operation of a system of records on individuals is required to accomplish an agency function:
Prescribing reference: FAR 24.104.
The official text, verbatim
As prescribed in 24.104 , insert the following clause in solicitations and contracts, when the design, development, or operation of a system of records on individuals is required to accomplish an agency function:
Privacy Act Notification (Apr 1984)
The Contractor will be required to design, develop, or operate a system of records on individuals, to accomplish an agency function subject to the Privacy Act of1974, Public Law93-579, December 31,1974 (5 U.S.C. 552a) and applicable agency regulations. Violation of the Act may involve the imposition of criminal penalties.
(End of clause)