You can argue the finer point of defining "enforcement" in a federal system (whether state [non-]cooperation counts seems to be your actual & rhetorical question), but if you're not asking this facetiously, the White House clearly has the latter in its sights; it's in fact the fist objective it lists in its page called
"Enforce Immigration Laws Across the United States"
STOP “SANCTUARY CITIES” States and localities that refuse to cooperate with Federal authorities [...]
STRENGTHEN IMMIGRATION LAW ENFORCEMENT: Hiring an additional 10,000 ICE officers and 300 Federal prosecutors [...]
Here's an example of self-declared selective enforcement at local level (from Oct last year, so it should be relevant enough):
As officials unveiled a series of public service announcements on the importance of reporting crime, Orange County Undersheriff Don Barnes, who is running to be elected sheriff next month, assured residents Tuesday that his agency is not enforcing immigration laws.
The television public service announcement are meant to assure immigrants living in the country illegally that law enforcement will not arrest them if they report a crime or are victims of a crime.
"There is some rhetoric in local media that law enforcement" has been enforcing immigration laws, Barnes said at a news conference at the Mexican consulate's office in Santa Ana.
"That is not true," he added. "We do not enforce immigration law. We have never enforced immigration laws and we will not enforce immigration law on the street level."
Barnes said any indication otherwise "sends a dangerous message to the community and erodes trust" in law enforcement.
"We don't ask for immigration status" when responding to any reports of a crime, Barnes said.
"If you call for help we will be there for you," he said.
"We will not be subjecting anyone to any inquiry regarding your immigration status."
Barnes, however, acknowledged that his department has cooperated with federal immigration authorities when it comes to transferring some Orange County jail inmates to a federal facility for immigration enforcement.
But Barnes pointed out that of 58,129 inmates booked last year only 580 were released to Immigration and Customs Enforcement officials.
"Those handed over to ICE are serious offenders."
And a similar one:
Following statements made by the President-Elect and various other governmental officials concerning the enforcement of immigration laws, there have been an increasing number of questions raised regarding the Santa Monica Police Department's official position on immigration enforcement.
The Santa Monica Police Department (SMPD) has a decades old policy and practice of leaving the enforcement of immigration violations to the federal authorities. This policy has been in effect because, similar to other municipal law enforcement agencies, we are sensitive to the principle that effective law enforcement depends on a high degree of cooperation between the SMPD and the public we serve. SMPD officers understand that the U.S. Constitution guarantees equal protection to all persons within its jurisdiction, in this case, to include the City of Santa Monica. Furthermore, we understand that all individuals, regardless of their immigration status, must feel secure in the knowledge that contacting law enforcement will not make them vulnerable to deportation. One of the primary reasons SMPD does not enforce immigration violations is to encourage those undocumented community members who are crime victims or witnesses to feel more at ease when coming forward to report occurrences of crime to the police. By not enforcing the immigration laws, SMPD encourages willing participation in our community safety efforts by those who might not otherwise come forward for fear of having to disclose their immigration status and risk deportation. As has traditionally been the case and will continue to be, SMPD officers do not inquire as to a person’s documented status; the only exception to this practice is if the person’s immigration status is materially relevant to another criminal offense or investigation, such as human trafficking, terrorist threats, etc.
And for some background:
In 1996, Congress authorized the attorney general to make agreements with state and local governments permitting them to enforce immigration laws. But before Sept. 11, lack of state and local interest, coupled with opposition by pro-immigrant and business groups, kept agreements from being struck.
Since the federal government can't just fire the local and state officials but has to make deals (arrangements) with them, you can see why this is not a simple "you're fired" matter. Trump tried to put financial pressure on "sanctuary cities" as they came to be known, but eventually gave up after legal challenges:
A judge in one of the cases against the Justice Department also said that local jurisdictions' decision not to assist federal immigration officers is not an obstacle to that immigration enforcement effort. Overall, courts have said that the administration cannot place conditions on the grants.
And since the confusion seems to revolve around the word "obligation" in the OP's formulation, I'll recall that selective enforcement of laws is legal in the United States.
Although the present administration has chosen to take a hardline approach on immigration enforcement, it can only do so (by itself) at federal level (via executive orders etc.) The discussion on selective enforcement is nothing new, since most previous administrations were less hardline, e.g. a 2014 article discussed:
Would President Barack Obama, by refusing to enforce the immigration laws against millions of undocumented immigrants, be engaging in “domestic Caesarism,” as Ross Douthat charges? The New York Times columnist thinks that “the granting of temporary legal status” violates the Constitution because Congress has itself refused to issue amnesties or offer a path to citizenship. [...]
Millions of illegal immigrants have lived in the United States for decades, under a semi-official policy that allows them to stay as long as they don’t commit serious crimes—and that, in many cases, allows them to obtain drivers’ licenses. The main effect of Obama's proposal would be to officially recognize current practice. The president cannot suspend or change the law: When he leaves office, the law will remain the same as it was, and the next president will be free to enforce it or not.
The executive branch spends a lot of time not enforcing laws. Congress has illegalized an enormous amount of activity without giving the president the resources to enforce the laws, so the executive has no choice but to make a list of priorities and devote its attention to law violations that, in its opinion, are the most serious. Thus, the IRS doesn’t audit paupers very often. The Justice Department ignores a lot of anticompetitive behavior that might raise prices a bit but not much. The DEA focuses on criminal syndicates rather than ordinary drug users, although both violate federal law. And so on.
Nearly all of this non-enforcement takes place with implicit congressional acquiescence; once in a while, Congress complains because the president’s priorities are not the same as its own. But the president has no obligation to listen to these complaints. [...] This is the separation of powers. People like Douthat wrongly think that separation of powers means that the president must do what Congress decides. That’s not the principle of separation of powers; that’s the principle of legislative supremacy, embodied in parliamentary systems like Britain’s, which America's founders rejected.
And neither do the local (or state) governments have to listen to moans from Congressmen, as far as I know. Federalism and all that.
Presumably Congress could pass a law allowing the Federal Government to pursue or punish local-government non-enforcers in some way. But until and unless that happens (on immigration matters)... there's no obligation for the local governments to enforce immigration law. Which doesn't mean however that these locals are enforcing the immigration laws.
The situation of marijuana laws in the US provides a pretty good analogy:
“that state decriminalization and even authorization of marijuana possession and use do not necessarily conflict with the federal criminal prohibition. If state officers were in fact responsible for enforcing federal law, or if the federal statute could and did require that state laws include comparable prohibitions, then a different situation would present itself. But state officers cannot be compelled to enforce federal law, and the federal statute does not require—and likely could not mandate—that state laws contain comparable marijuana prohibitions.”
“It’s not that the state laws are attempting to nullify the Controlled Substances Act. The federal statute still applies to the activities that it specifies, is still enforceable by federal agents, and can still lead to federal prosecutions, convictions, and sentences. It is simply that the states themselves are no longer criminalizing some of these activities. And while that may disappoint federal expectations, and even conflict philosophically with the federal outlook, that circumstance by itself does not give rise to preemption under the Supremacy Clause.”
