Can a signal be sent and received in the same point? What are the limitations? I'm wondering about this scenario:
START: $B$ and $A$ moving to the right with speed $u$
$A$ present in point $X$ 
Signal $S$ with speed $v'$ sent by $A$ in point $X$ towards $B$
$\require{AMScd}$
\begin{CD}
    B @<S<< A\\
    @.  @|\\
     @. X
\end{CD}
END: $B$ reaches $X$ and receives the signal.Signal $S$ with speed $v'$ received by $B$ in point $X$
\begin{CD}
    B @<S<< A\\
    @|  @.\\
     X @.
\end{CD}
The scenario involves three speeds:


*

*$u$ - the speed of the inertial system moving right

*$v'$ - the speed of the signal as perceived by A and B, it should be so, that the signal reaches B in the same point X in the outer system

*$v$ - the speed of the signal as seen from point X, it should be $0$



My questions:

*Can such a signal be sent at all?

*Could it be a light signal? Any other limitations?

*Could you give an example with explanation?
Please note it's tagged popular-science, so answer as simply as possible, but not more.
 A: Non-relativistic scenario  
If the speed of the signal is $v'$ according to the observers $A$ and $B$ moving with speed $u$ to the right then the speed of the signal observed from the stationary point $X$ will be the difference between the two velocities:
$$u-v'=v.$$
Where $v$ is the velocity of the signal observed from point $X$. If the signal's speed seems to be zero it means that
$$v'=u.$$
This can occur if the signal is carried by a piece of paper and it is thrown form point $A$ to point $B$ in space where no gravitation or air resistance has effects on their movements. This version is definitely possible to do. But let's get started, we want to know what happens when the signal's speed is the speed of light! 
Relativistic scenario 
We need to start with the most fundamental postulate of relativity: the speed of light is observed to be the same from any point of view. This means that it does not matter how fast I am moving, if I shine a laser in the opposite direction I am moving it will travel with the speed of light and won't "loose" even a bit of it's velocity even if I am moving with 99.9999999% the speed of light.(More on special relativity here and here) This means this experiment can not be done with light signals because nothing can slow down light! 
Final conclusion 
The experiment can not be done with any signals which are travelling with the speed of light(radio waves,visible light and any kind of electromagnetic waves)But it could be done with moving objects (even atoms or anything what has mass). I hope my answer was useful :)
