In general, statistical dependence is not transitive. If $Y$ and $X_{1}$ are dependent, and $Y$ and $X_{2}$ are dependent, then $X_{1}$ and $X_{2}$ are NOT necessarily dependent.

However, in some cases they CAN be dependent. A question which occurred to me earlier this week on which I have not been able to make some headway, is: are there any random variables $Y$ such that any variables dependent on $Y$ must be dependent on each other?

Related is the idea of decomposability (whether the random variable can be expressed as the sum of independent random variables), but I was unable to find anything on this more general problem.

The simplest case which occurs to me is a Bernoulli random variable, but I have neither been able to prove it nor offer a counterexample. If someone could shed light on the issue, it would be much appreciated.