Well, your friend is correct that under the present system of government-funded police/military/law enforcement, Person 1 (the billionaire) doesn’t have to go to the bother of having mercenaries available at his personal beck and call, because the State can just have Person 1’s land seized under Eminent Domain (see Kelo), and the State’s police + law enforcement + courts will enforce this redefinition of property rights (the State is always redefining property rights in favor of itself and it’s hangers-on). Indeed under present law as defined by the State, Person 1 may “own” the house but not the mineral rights under it anyhow, these get assigned by the State to Persons 2 of it’s choosing [this includes the State preventing Person 2, if Person 2 “owns” the mineral or other resource rights, from accessing them or selling access to them, because the State uses a corrupt version of “environmental” laws to prevent many people from accessing/selling/profiting from the resources they theoretically “own” but the state, with it’s police-law-forces-etc that your friend points to as “protecting” Person 2’s property, protects him from either using them himself or selling their use to another person. The point being: in more instances than most people are aware of, the State is not protecting Person 2’s property rights against the rich and influential, but rather allowing the fashionably rich & influential to control public policy in a way that deprives Person 2 of the property rights they are ostensibly “protecting.” Five will get you 10 that your friend approves of, or at least doesn’t object to, this seizure of Person 2’s property rights – under the grounds that, after all, “we” can’t just have everyone drilling oil on “their” land – after all, “they didn’t build that!” – “society” has claims too!]
Ok, ok, I’ll stop being droll, but the Anarcho-Capitalist solution to this is that Person 2 pays “insurance” to a private protection agency, the size of which would probably be of a scale that it would have sufficient resources to back the legitimate claims of its members such that it would be easier/more efficient for Person 1 to simply negotiate in good faith with Person 2 after all.
That is, these private protection firms would have police + law enforcement on the payroll of a power (if not size in number of personnel, because of being run more efficiently) to enforce the legitimate property rights of all their subscribers, to include Person 2.
Anyhow, once again a true AnCap (I remain a minarchist, but I do strive to understand – and want to agree with – the AnCap position) could probably provide a better answer than I just did, but that’s the basic outline.