Two masses in empty space Consider two equal point masses which are rotating circular arround each other in empty space for ever (radiation effects are ignored).
Let an observer be at the center of mass of the system. Then, there seem to be two possibilities:


*

*The masses are rotating relativ to the observer.

*The masses are not rotating with respect to the observer. 


In the second case, the observer implies that both masses should come closer (because of the absense of radial forces) and reach each other after a finite time. Shouldn't the result whether the masses reach each other or not be independent of the observer?
 A: If I have correctly understood your question, the answer is: yes, the distance between the two masses, being a scalar quantity, does not depend on the particular observer (given that we can neglect relativity effects, which are not the point of your question).
Your analysis of the second statement is therefore incorrect: a situation in which the masses are not rotating with respect to the observer is only possible if the observer itself is in the frame of reference of the point masses; in this frame, both masses have zero velocity. This is, however, a non inertial reference frame: it means that inertial forces (or apparent forces) act on the point masses; in particular they are subject to the so-called centrifugal force $\textbf{F}_c = m \frac{v^2}{r^2} \textbf{r}$ (where $v$ is the tangential speed measured in the inertial frame at rest w.r.t. the center of mass of the system).
This force is what "prevents" the masses from collapsing at their center of mass, in their own reference frame.

EDIT (in response to the comments by David White and OP)
In order to make a measurement, an observer does not need anything to measure against (as the comment by David White suggested).  
A point mass moves in circular motion around the center of the cirumference, which is a geometrical point, and there is no need for a background in order to measure physical quantities such as the radius of the circumference or the speed of the particle. 
Your comments may be summed up in the following question: "How does an observer, placed in the center of mass of the star system, infer that the stars are rotating, if the observer itself is at rest w.r.t them, given that there is nothing else whatsoever in the Universe apart from the two stars?"
The answer is with proper measuring techniques. For example a mass attached to a spring could be used: it would feel a centrifugal force (see above) because the frame is not inertial.  
What I am saying is that the effects of the non-inertiality of the frame such as centrifugal forces are intrinsic and self sustaining.
Therefore you do not need to see a background of galaxies circle above your head in order to realise that you are moving in some way. If you reference frame is non-inertial, there are ways to measure it that are independent from the exterior.
A: Consider the reality that the observer, in the second case, would be able to detect or feel his own rotation (his arms would tend to fly outward) and conclude that the masses are revolving around their center of mass even though it visually appears they are motionless.
