There are union contractors that hire clean-up crews of illegals just like open shop contractors.
Most contractors can't tell when an immigrant worker being hired is illegal or legal. There wasn't any way to check SSN numbers by web until just recently. Most small firms don't yet know how it works. In fact, the biggest number of Civil Rights complaints to contractors is on the issue of documentation or immigrant status impacting employment. So many construction contractors are damned if they avoid the hispanics and damned if they don't.
The issue here is that the leftists in the Senate are trying to apply the Davis-Bacon act to private work which will drive up the price of all construction work including homes by a huge percentage and using this bill as a wedge tool to do it.