Which one is inertial frame? I'd like to ask a question which made me confused. I think that, e.g., a man falling to earth could think himself as stationary and the Earth as accelerating towards him. But I, as an observer, I think that man is falling to the Earth with 'no doubt'. 
So, in Relativity it is crucial to determine which system is an inertial frame. How should I decide which one is accelerating, the Earth or the man? What if I and the Earth would be accelerating with the same velocity towards the man, while I think myself stationary? 
If there is a question like mine sorry for duplicating it. I looked if there is such a question but maybe I missed it. 
 A: 
I think that, e.g., a man falling to earth could think himself as
  stationary and the Earth as accelerating towards him

The man is stationary, with respect to himself, and assuming he is falling freely, he has no proper acceleration; his accelerometer reads zero.

But I, as an observer, I think that man is falling to the Earth with
  'no doubt'.

It's not clear what you mean.  Are you an accelerated observer at rest on the surface of the Earth?  If so, then you are clearly the one accelerating according to an accelerometer attached to you; your accelerometer reads $1g$.
So, according to your accelerometer, you are 'no doubt' accelerating.
To get to the bottom of this, we must distinguish between proper acceleration, which is acceleration 'no doubt', and coordinate acceleration which is relative.

How should I decide which one is accelerating, the Earth or the man?

If you're at rest on the surface of the Earth, you would decide that


*

*you have proper acceleration while the man does not

*the man has coordinate acceleration while you do not

A: On earth, you are definitely not in an inertial frame. In a truly inertial frame, when you just let go of an object, it will stay where it is. Because you reference frame is inertial; the frame does not accelerate, the object does not accelerate, so it stays put. By "let go", we essentially mean that there is no external force acting on the object.
When you let go of an object on the Earth, it will "fall down". Because the frame you are in (which you call "stationary") is constantly accelerating upwards. Let go of an object, and you accelerate up, the object stays behind, and hits the Earth.
Of course, saying "zero external force" treats the force of gravity as equivalent to the force felt in an accelerating frame; this is called the "equivalence principle". So, in the framework of general relativity, gravity is not treated as an external force; it is said to bend space-time. So, the Earth, with its big mass, constantly sucks the space dimensions into it. You need to accelerate just to keep from getting sucked in -- and that force is supplied (given you are standing up) by the contact forces (which are essentially electromagnetic in nature) between your feet and the ground. All in all, standing on the Earth is definitely not an inertial frame. (Did I say that already? Yes I did.)
The guy in outer space is right. If he lets go of an object, it will stay fixed relative to him. And yes, the Earth is accelerating towards him! (Aaarrghh!)
A: One way to test whether a frame of reference is inertial in Newtonian mechanics is based on Newton's first law: Determine whether a particle on which the net real force is zero moves along a straight line path. If it does, you might have an inertial frame (but you need to test motions in multiple directions). An alternate definition, based on Newton's second law: Determine whether you have to create fictitious forces so as to make Newton's second law appear to hold. If you don't, you have an inertial frame.
Note the key qualifiers "real" and "fictitious" used on the word "force". One big difference between inertial frames in Newtonian mechanics versus those in general relativity is how acceleration due to gravitation is treated. Gravitation is a real force in Newtonian mechanics, but a fictitious force in general relativity. A person orbiting a planet with no external forces other than gravity is accelerating (not inertial) in Newtonian mechanics. This person is moving inertially in general relativity. 
Another key difference between inertial frames in Newtonian mechanics versus those in general relativity is that inertial frames are universal in Newtonian mechanics, but local in general relativity. Suppose our orbiting person sees another spacecraft in a different orbit. The people on each spacecraft think they are at the center of an inertial frame. Yet each sees the other as accelerating. How can two inertial frames be accelerating with respect to one another? The answer is that inertial frames in general relativity are local. Just because everything in that astronaut's immediate surrounds appear to be behaving inertially doesn't mean that the astronaut should expect freefalling elsewhere to not have an observable acceleration.
Finally, what about the person on the ground? When he drops a pebble, it accelerates Earthward. Things don't behave inertially even in his immediate surrounds. From the perspective of general relativity, our earthbound observer is not inertial. 
General relativity provides yet another way to test whether a frame is locally inertial: What does an accelerometer report? An ideal accelerometer measures acceleration with respect to a local inertial frame. If an accelerometer reports a non-zero value, the frame isn't inertial. An accelerometer on the surface of the Earth reports an acceleration of 1 g, directed upward. This is not an inertial frame, from the perspective of general relativity.
A: 
How should I decide which one is accelerating, the Earth or the man?

Here is pre-relativistic view on this issue.
In Newton's view, acceleration that appears in his laws of motion has definite absolute value, meaning it is referred to particular reference frame called absolute space(frame). It is an hypothetical thing (I imagine something like rigid set of points) pervading all the Universe in which laws of mechanics (3 laws of motion + the law of gravity) are valid.
However, the concept of absolute frame has bothered people, perhaps also because it suggests the picture of one frame while all its characteristic features - validity of Newton's laws - are shared (in Newton's view) with infinity of reference frames that move rectilinearly with respect to it. Absolute space was never identified with any body. It remained hypothetical.
So the notion of inertial frame is often preferred - it is any frame in which Newton's 3 laws of motion are valid.
With this notion, one possible answer to your question is then: find some reference frame in which those 3 laws are valid and measure/calculate accelerations of bodies with respect to this frame!
Is the astronaut an inertial frame in this meaning? Let's see. The astronaut knows there is force acting on him due to the Earth but he does not move in his own frame. Oops. Similarly for the man on the Earth, he knows he should be attracted to the astronaut but sees no acceleration of his body either. So neither of these two frames obeys the requirements imposed on the notion of inertial frame perfectly.
However, if we assume that Earth is inertial and calculate the motion of the astronaut into future or past, we will get very good answers close to his actual motion, while if we assume astronaut is inertial and calculate the motion of the Earth we will get ridiculously inaccurate results. So even if neither of the two is absolutely inertial, Earth is much closer.
Another example, motion of the Earth and the Sun is well described by Newton's mechanics when Sun is assumed inertial. If Earth is assumed inertial,it will not work well for the motion of the Sun - the value of calculated gravitational acceleration of the Sun due to gravity of the Earth is much too weak to explain Sun's acceleration in the frame centered in the Earth.
Thus it looks like the heaviest body in the system is a good choice for the reference frame in which the 4 laws are going to be applied. If you want to make as accurate predictions/retrodictions as possible, I would use the 4 laws in the frame that


*

*is centered at the center of mass of the system

*has axes that do not move with respect to distant stars.
It won't be absolutely inertial, but should be close enough (providing other bodies are much farther).
