Inspired by the question: "How would the world governments react if the supernatural was discovered?", I have a question that is a variant on this theme:
Thousands of years ago, the rational and magical worlds split as the result of a magical war, the consequences of which split reality itself, with most of the things that were magical ending up over there in the magical realm, and most of the rational, explicable things ending up over here in the rational realm. The rift between the magical and the rational deepened until the two halves of Earth's reality were so 'far' apart that communication between them became vanishingly rare, limited for the most part to chance accidents resulting in a person slipping from one reality to the other, and the actions of powerful magical beings who - for the most part - aren't interested in the deeply un-magical rational half of reality.
However, the two halves of reality were always meant to be one, and a goddess from the magical half of Earth's reality has decided to work toward bringing Earth's magical and rational realms back together.
In Earth's rational realm, things have progressed as we have experienced here on our earth. Magical events and creatures have been dismissed as myth and legend, and no scientific proof that they ever really existed has ever been found.
However, in the magical half of Earth's reality, some of humanity has escaped the bounds of Earth and has spread out amongst the stars along a magical system of marginally-breathable air-filled 'tunnels' at about 0.35 ATM pressure, known as the Gyre, that forms a network connecting stars with planets that have oxygen-containing atmospheres, that is shorter on the inside than outside. The reduction of real-space distance in the gyre is approximately:
Gyre Distance (km) = 2.5 × π√[Normal Space Distance (km)]
So, as an example, a distance of 50 lightyears between inhabited systems (4.73 x10^14 km) would be reduced to 2.5 times the PIth root of that distance, or about 117,000 km of air-filled 'tunnel'. This represents the typical minimum gyre-distance between
worlds gyre waypoints, several of which need to be passed to travel to another world.
So, given that background, we have a modern-day Earth, almost completely unsuspecting of the magical realm that is so near yet so far away, until one day magical, inexplicable things start happening, at first small, but quickly ramping up to things such as a whole city-state on the top of a mountain over three kilometres high appearing with the dawn, remaining for most of the day, then disappearing and appearing elsewhere at that new place's dawn, or people who claim to come from other worlds showing up in hundred-metre-wide flying saucers that mass about the same as a World-War-II battleship, hovering over major population centres and requesting to open trade relations.
If any government tries to shoot the flying saucers down, the saucers are also armoured like WWII battleships (or better), and anti-aircraft missiles would have little effect on them - it would take an all-up anti-ship missile or a close-ranged nuke to have any significant effect on them, and these flying saucers have the ability to outrun (top-speed in-atmosphere of 7,000+ kph) and out-manoeuvre (4g turns) such missiles, and are also armed with a variety of weapons including lasers powerful enough to bring down any aircraft and seriously inconvenience a modern naval ship. If a flying saucer was to be brought down, their power sources, drives and much of their other equipment cannot be duplicated in a non-magical environment such as our rational realm.
Conversely, if any governments attempt communication with the saucers' crews, they would be readily told that there are a dozen or more worlds accessible via the Gyre that are inhabited by humans or human variants that these travellers are attempting to bring back into contact with one another via peaceful trade relations, in addition to which there are many worlds with alien life, some of which is sentient and also travels the gyre.
If people from the Rational world were even able to find and enter the gyre (it's pretty simple - go to about 10,000m and fly directly away from the sun at 40-85 kph in a place that is gyre-accessible, which at present eliminates most if not all of the Rational realm), I'm not sure that any of our aircraft have the speed or endurance required to make the 100,000km+ journey through the gyre to the nearest other world - the flying saucers use a couple of varieties of magical-phlebotinium-fuelled atomic power.
The relocating mountain is the seat of power of the winter goddess' avatar, and is itself set up as a trading society, trading magical goods and educational services for food and other goods that require large areas of non-winter-bound land to produce, and is itself heavily fortified to discourage the communities it visits from deciding to raid rather than trade. This mountain and its goddess are unrelated to the saucer-people, but they know of and trade with one-another.
So, how would our world's governments and communities react to these revelations?
There's no chance of hushing them up, given all the civilians who will see 'impossible', magical things. While our technology is in many ways advanced over that of these magical people, at least some of them have combined the technology they have with magic, and in many ways we are playing second-fiddle to these travellers with their magical aircraft and the moveable mountain that we can't duplicate or emulate - yet.
EDIT In response to an answer that posited that a war would be inevitable:
The Gyre consists of air-filled tunnels with insubstantial walls that range from 50 to 1000 kilometres wide, at an atmospheric pressure of 0.35 ATM, equivalent to a terrestrial altitude of 9-10 thousand metres, though the atmosphere is about 38% Oxygen, 55% Nitrogen and 6% Helium. While the net air speed within a gyre stream is on the order of 5kph, each stream has several air currents that flow in either direction at up to 2,000kph, that contain particulates at their cores, some pieces quite large, as well as frequent lightning activity - there's a reason that gyre-capable aircraft are heavily armoured - being hit by a stray rock travelling at up to 12,000kph relative to your aircraft is no joke.
It is possible to 'fall out' of the Gyre into interstellar space, from where return is effectively impossible. The gyre streams are hundreds of thousands of kilometres long; to reach the nearest gyre-accessible world nearest to Earth would be a journey of around 660,000 kilometres - not too difficult in space, but when flying through air, that's impossible without either atomic power, or the willingness to take weeks to months, depending on how close to the dangerous high-speed currents you're willing to get. Our aircraft just aren't made to have that kind of endurance.
Then there's the problem of navigation - the saucer people use a combination of inertial and spectrographic methods, without which the gyre is a confusing maze of tunnels joining light and dark nodes.
The saucer people don't just have flying saucers with hulls with the equivalent of over 2.6m of battleship armour in every aspect, (which can carry up to 78,000 metric tons of ordnance, war material and personnel if necessary). They have access to thousands of mecha: large mecha armoured about as well as a modern main battle tank, and smaller ones that are still armoured as well as a light tank. They have atomic-powered fighter aircraft with reactionless thrusters that can pull 14+g manoeuvres without inconveniencing the pilot, armed with the aforementioned lasers plus missiles with atomic engines and all-aspect homing - they don't lose target lock, you have to actively break the (magical) lock, or kill them. They have tractor/repulsor beams that could lift a modern cruiser out of the water or capsize an aircraft carrier. Their ground troops are protected by body armour that can stop almost any pistol bullet, and provides decent protection against rifle ball ammunition up to .30", even directly to the visor of the helmet - it'd take a .50"+ bullet or an AP round to seriously inconvenience the wearer.
On the other hand, the saucer people don't want a war - they're traders by inclination, but from a military background, so they'll defend themselves if pushed to it.
The Winter Goddess doesn't want a war either, but if pushed to it, she has enough magic to teleport a whole 3km-high mountain on a daily basis, or turn an enemy army into a collection of ice-rimed statues.
Then, while there are multiple magical realms local to Earth, their populations are for the most part incapable of dealing with a modern military, but they also have magi, some of whom specialise in fighting, and even one could seriously inconvenience a modern technological army.
I'd anticipate that any war between the magical and rational realms would be bloody and inconclusive, with heavy losses on both sides, and would most likely result in Rational Earth being abandoned to its fate by the magical peoples, knowing that there was a whole galaxy out there, but denied access to it in the short-to-medium term.