Let $E$ and $E'$ be nonisogenous elliptic curves over a field $k$ (characteristic 0) such that $Gal(k(E[p^{\infty}])/k)=Gal(k(E'[p^{\infty}])/k) = SL_2(\mathbb{Z}_p)$ with $p \geq 5$ (where $E[p^{\infty}]$ is the set of $p^n$ torsion points of $E$ for all $n$). Then is it true that $k(E[p^{\infty}])\cap k(E'[p^{\infty}]) = k$, or can someone provide a counterexample?

Since both fields $K(E_{l^\infty})$ and $K(E'_{l^\infty})$ contain the $l$adic cyclotomic extension of $K$, your expectation cannot hold. However, this is almost the only obstruction. In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259331 (1972), JP. Serre proved the following Theorem (Theorem 6$''$, p. 325).
(By Faltings, hypothesis (ii) is equivalent to the one given by Serre.) 


By the way, I think that under your hypotheses, your question is really about group theory, not about algebraic geometry. Namely: the action of Galois on E[p^infty] x E'[p^infty] gives you a homomorphism G_K > SL_2(Z_p) x SL_2(Z_p). Call the image H. By your hypothesis, H projects surjectively onto both copies of SL_2(Z_p). You also know that H is not contained in any conjugate of the diagonal (if it were, E[p^infty] and E'[p^infty] would be isomorphic Galois representations and I'm presuming you're in a situation where Faltings rules that out  you'd better be, if you want an affirmative answer to your question.) Now what you have to prove is that a subgroup of SL_2(Z_p) x SL_2(Z_p) which projects surjectively onto each direct summand and which is not conjugate to a subgroup of the diagonal must be finiteindex in SL_2(Z_p) x SL_2(Z_p). This is true for SL_2(F_p) by Hall's lemma and I think you can induct from there (but didn't think about it carefully.) 

