The question is just this:
Can a private citizen of a given country host a military base of another country on his private property?
This might be especially important if this property is close to the borders/oceans?
(e.g. A US citizen with large private property on Canada border hosts a Canadian Military Base)
No
A private citizen does not have the right to issue visas, or other travel documents to citizen soldiers of a foreign country.
Private citizens are bound by laws. They can't have military weaponry "on the farm"
Private citizens are not permitted to negotiate with foreign powers as if they were the government.
So the whole idea is a total non-starter.
Even though Liberal nations have the concept of private property, all territory in the state is fundamentally 'owned' by the state. Private property means that a given individual is given the right to use some property as he sees fit and without government interference, on the condition that the property is not used for activities contrary to the state. One cannot use private property to run an illicit drug lab, enslave workers, set up a murder program, develop an insurgency force, or do anything that goes against the laws and interests of the state as a whole.
Hosting a military force (foreign or domestic) without the consent and approval of the state is intrinsically against the interests of the state. There are certainly allowable cases — e.g, if a private individual hosts a foreign military group for some limited training period, or for some public performance — but long-term hosting of foreign military would be tantamount to an invasion, and the state would not view it kindly. It's a quick way top get your property seized and yourself in jail.
Having heavy weaponry is illegal in most countries, even if you store it on private property. So no advanced weapons, which military bases often imply. Otherwise, you may be tried for weapon smuggling.
Military bases which only have permitted/light weapons are called "training camps" and are sometimes tolerated even when nominally held by private persons or organizations. You should make sure to only train allies of your country, though. Or that would be, at worst, treason.
It is possible. But the government of the country the base will be located in has to either agree to it, or be powerless to stop it. In both cases, it would mean that government power is tenuous, and the country is not stable; this means that the land owners are likely to not be happy with the deals they get.
A recent example are the (former) US military bases in Afghanistan. This 2020 NYT article mostly deals with lands seized against their owners' will, but it does mention that in some cases, temporary lease agreements were drafted - i.e. almost exactly the situation you are asking about. Almost - because the owners did not really make a chice to host the base; rather the choice was "the base will be built whether you agree or not; you can agree to this deal and get at least something out of it, or you can protest and have your land seized". Needless to say, it was only made possible because the official Afghan government had absolutely zero concern with American troops on their soil and every reason to accomodate them in any way possible. THey were unable to establish a military presence in their own land needed to control it, so they had to present USA troops with many base locations, often in places where land was not government-owned.
On the other hand, any government secure and stable enough will want to a) keep the amount of foreign military bases to a minimum, and b) exert as much control over foreign troops on its soil as possible. As there is no need to establish a strong network of foreign military presence in strategic locations (which was the case in Afghanistan, since the government relied on foreign military help to stay in power), the placement of the bases can be selected based on what is more convenient to the host country. So the bases are usually on government-owned land; in many cases - these are pre-existing military bases leased temporarily directly from one government to another. So there is no need to permit private landowners to host a base on their property.
There is, of course, a simple scenario where both governments agree about the military base and its exact location (be it because it is viewed as mutually beneficial or because one of the governments doesn't really have a say).
Then, because the location happens to be on a private land, the "host" government buys, rents or sequesters the land from its owner for the duration of the base existence.
Most countries have legal provisions allowing the government to use private property (not only land) for defense purposes, with or without compensation for the owner, with or without the owner approval.
All 4 combinations in regard to owner control and compensation do exist even in the most liberal countries. Of course, the unpleasant ones are rarely invoked in peace time.
No.
State sovereignty is protected by international law. This principle specifies that only the government of a country has the power to give permission for another country to build military base on its territory.
A private citizen does not have the right to give such a permission (unless he or she is a constitutional monarch who has to formally sign the document on the instruction of their government). If someone actually tries to do that, the foreign state being offered the land would rightfully refuse or risk becoming an international pariah.
Possibly under a very narrow set of circumstances. A private citizen who owns a large area of land in the United States can lease that area to the US government. The US government can then use that land for the period of the lease to house armed forces of a different state.
As another answer already mentioned, the US government would have to issue visas to the hosted servicemen. The US may also make those visas restricted to the area of land where the base would be +/- some surrounding area and a travel corridor to/from their home country.
Transactions with foreign militaries are often (maybe even usually) billed to the US military and the US military recoups the money from the foreign military. At lease, that has been the case in the only instances that I came across. This takes away the need to deal with any export restrictions, sanctions, etc.
The reason a private citizen may wish to lease the land rather than outright sell it is that it may appreciate in value significantly over a long period of time that a base would be hosted. And if there is a need to extend the lease, that would put them at a significant negotiating advantage.
If the deployment is known to be time-limited and the land is significantly better than other land useful for that purpose, it would also give the owner a significant negotiating advantage.
I don't believe (although I am not certain) that Eminent Domain can be used to disposes someone of land for temporary use. In the only cases that I know of, land taken over, through Eminent Domain, has been for permanent use (construction of roads or some permanent structures). Which would make it more difficult for the government to convince the court that Eminent Domain can be used to take over land for temporary use.
In principle though, it's very common for the government to house its offices and facilities in buildings which are privately owned and privately managed. Doing this for the benefit of a military of a friendly nation for the purposes of advancing some US national security goals seems like an ordinary use of the US resources.
