IMO, this is a job for which the unorganized militia is ideally suited. Put up a bounty for illegals turned in with substantial penalties for false arrest or harassment. Put up a web page instructing the potential "bounty hunter" in the proper investigation, evidence collection, arrest, and detainment procedures along with all the applicable statutory law.
There would be companies set up in weeks dedicated to the purpose. There would be traffic jams at the borders. It's not that hard.
As to its constitutionality, this method is effectively a letter of marque, while the professional police state we've built in the form of ICE is clearly the standing army the Founders feared. The Constitution is OUR document, OUR Supreme Law, and not just for the government. When are we going to get it?
Yup. And worse, it's now on THEIR side. They've become professional aiders and abetters!
The police state has been told that enforcing laws against foreigners is not allowed. Only against us, because we "agree" to have the laws enforced on us.