Dozens of migrants working illegally at the Byron hamburger restaurant chain were caught by immigration officials, in welcome news for the new Home Secretary Amber Rudd. She will hope these headlines continue.

The previous government under David Cameron made it more difficult for migrants to work illegally. They did this by requiring new checks for work, renting property from a private landlord, opening a bank account and acquiring a drivers' licence. Applicants have to prove their lawful residence. The plan was to make everyday living in the UK too difficult for illegal migrants. Eventually, they would be found out and could be subsequently removed.

The logic behind this is that most illegal migrants are discovered after they cross the border – not at the border itself. A travel or work visa may expire, but the government lacks the tools to know where these people are. Its plan is to expose them when they engage in everyday tasks like renting a home, using the bank or driving a car.

Some industries are targeted by immigration officials more than others. One area receiving significant attention is the restaurant industry, especially takeaways, where illegal working is rife. The government has been increasing its searches of businesses where illegal working has been a problem historically, given the limited resources for such work.

Yet it remains unclear how much the government's tougher residency checks contributed to exposing the Byron burger chain. Byron had already checked the residency papers for its staff, including the 35 people found to be working illegally from countries like Albania, Brazil, Egypt and Nepal.

The checks were lawful, but the documents provided were not. We have now learned that these employees had submitted false documents.

This poses a real challenge for the Home Office. It has effectively devolved much of its immigration enforcement efforts to us. It falls to employers, private landlords, banks, universities and others to do the document checking. Many of us are now de facto border agents.

The problem is that we may act like border agents, but we are not them: we lack their training, their experience and resources to make the full checks available to the Home Office. They are better placed to uncover false papers, yet it is us who do this important work.

I strongly suspect the "intelligence" that led to this expose was a tip-off from the local community and that this brought about the result, and not any of the government's recent measures to tackle illegal working. This does not mean these measures aren't worth having, but that immigration officials are in a difficult position trying to tackle the known unknown of illegal workers in our midst armed with a lack of resources and people on the frontline to provide the checks necessary.

This highlights the need for greater public investment in immigration officials to carry out proper checks to enforce existing regulations and keep the community safe. The public need action backed by resources, and not gimmicks that may come to nothing.

Prime Minister Theresa May is in a strong position in her new role to get this right and help make the Home Office fit for purpose. The public is concerned about immigration more than any other issue, precisely because it lacks confidence in the government's ability to exercise control over our borders. Improving efforts to tackle illegal working can both promote protecting the public and rights in the workplace, while getting the balance right on migration. She should act decisively sooner, rather than later.

Thom Brooks is Professor of Law and Government & Head of Durham Law School at Durham University. His book Becoming British is published by Biteback. He tweets at @thom_brooks.