Is any one compact dimension for one particle the same as for another particle? In the 3+1 dimensions of everyday life and GR particles can share the same extended dimensions. Probably all particles share the same 3+1 dimensions.
In string theory compact dimensions seem to be limited to each individual particle. Is there any meaningful, or otherwise, sense in which the compact dimensions of one instance of a particle are shared with, or the same as, the compact dimensions of another instance of the same type, or a different type, of particle ?
Is there anyway that these compact dimensions interact ? If not, what purpose do they serve ?
 A: It seems that this question is asked because of difficulties in trying to visualize higher dimensions. 

Fig.1: The full manifold is here $\mathbb{R}^2\times S^2$.
I) Let us therefore for simplicity assume that the physical universe is just a 2D cylinder$^\dagger$ surface $\mathbb{R}\times S^1$. Imagine that there are only one large (uncompact) $x$-direction and one compact $y$-direction, where $y\sim y+2\pi R$ is periodic, and $R$ is small.
      ^ y
      |                
      |          .A <---image
 -----|---------------- y = 2 pi R-------------------
      |
      |                                    .B
      |
      |
      |          .A
 -----|------------------ y = 0------------------------> x
      |
      |                                    .B <---image

Fig.2: Two point particles $A$ and $B$ on a 2D cylinder. The two lines $y = 0$ and $y = 2 \pi R$ should be identified.
Let $A$ and $B$ be two point particles. $A$ and $B$ then live in the same uncompact $x$-direction and in the same compact $y$-direction. On the other hand, the positions $(x_A,y_A)$ and $(x_B,y_B)$ of the two points are not necessarily the same.
II) The above Section I is only meant as a guide to get the main idea. In string theory, we often assume that the full spacetime is topologically a direct product of a large 4D spacetime and a small compact space. In more elaborate models, this needs no longer be the case. For further information about string theory and extra dimensions, see also e.g. this and this Phys.SE posts. 
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$^\dagger$ Technically, we imagine the 2D cylinder surface as an abstract 2D manifold rather than an embedded 2D manifold (so that we don't have to introduce more than 2 dimensions).
A: Let me try to rephrase what you're asking. Suppose we have the usual spatial dimensions $x$, $y$ and $z$, and a compact spatial dimension $w$. Then can we have two particles simultaneously at positions:
$$ P_1 = (x, y, z, w) $$
and
$$ P_2 = (x, y, z, w + \delta w) $$
In other words the particles are at exactly the same position in the normal coordinates and differ only in being at different positions along the compact coordinate.
If this is a fair summary, then provided the size of the compact dimensions is much greater than the Planck length the answer is yes. This would be no different to two particles being at $(x, y, z)$ and $(x, y, z + \delta z)$.
However (I believe) the scale of the compact dimensions is normally considered to be around the string scale. If so then I would guess the question has no meaning because the point particle isn't a good description of how matter behaves at those scales.
