No. There is nothing called a private bill. They make laws for the government not themselves.
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Article I, section 7 mandates that all appropriation bills start in the House of Representatives. "All bills for raising revenue shall originate in the House of Representatives; but the Senate may propose or concur with amendments as on other Bills."
The congressional members that submit the final bills that authorize specific spending are the members of the House and Senate. They have budget committees that submit the bills.
The House of RepresentativesArticle I Section 7 Clause 1 states:"All Bills for raising Revenue shall originate in the House of Representatives; but the Senate may propose or concur with Amendments as on other Bills."
A private bill
The Parliamentarian of the United States Senate