Like many laws over the centuries they have been enforceable by both the federal and state governments.
The Runaway Slave act springs to mind. There were federal penalties applicable to local sheriffs who used to assist in enforcement by slave catchers. More recently we've had states busting dope peddlers (under state law) and handing them over to the federales for concurrent prosecution under federal law.
It's really not a revolutionary idea that states can assist in dealing with federal problems.
In reality the Constitution doesn't even mention IMMIGRATION. You might check what the USSC has said in the past, specifically that the Congressional power to regulate naturalization, in Article 1, Section 8, includes the power to regulate immigration (see, for example, Hampton v. Mow Sun Wong, 426 U.S. 88 [1976]).
So fine, the Feds can deputize whom they wish. Big hairy deal.