anti ice sign on a retail store

[SickOfTheOnePct]

8. The "randomly trolling part"...

...is where the loitering would come into play.

But if they have an immigration warrant for someone, it is completely lawful for them to follow that person into the public part of a business (or a park, etc.) and to arrest them. Or, to go into the business, say a restaurant, and wait for the employee to enter the public part of the restaurant and arrest them.