If two elliptic curves over $\mathbb{Q}$ are $\mathbb{Q}_p$-isogneous for almost all primes $p$, then they are $\mathbb{Q}$-isogenous.
This follows from the fact that they have the same number of $\mathbb{F}_p$-points for almost all $p$, hence their $L$-functions have the same local factors at all these $p$, therefore a combination of "multiplicity one" and Faltings' isogeny theorem implies that they are $\mathbb{Q}$-isogenous. Correct me if I'm wrong.
Here $\mathbb{Q}$ can be replaced by any number field $k$.
Question : Does the same argument work for any two abelian varieties $A$, $B$ over $k$ ? It should, since $H^i$ is $\wedge^i H^1$ for $i>0$.
If so, this explains why Poonen's abelian surfaces $A,B$ (everywhere locally isomorphic but not isomorphic) are $\mathbb{Q}$-isogenous.
Best Answer
Yes, the local-global principle for isogenies is valid for all abelian varieties over all number fields, as a consequence of Faltings' isogeny theorem [and, as Kevin Buzzard points out, of the semisimplicity of the Galois action on the Tate module, also proved by Faltings.]
The proof for abelian varieties is almost the same as that for elliptic curves. You just need to observe that if A, A' are two g-dimensional abelian varieties over F_q, then the following are equivalent:
(i) They are $F_q$-rationally isogenous.
(ii) They have the same characteristic polynomial of Frobenius.
(iii) For all $1 \leq i \leq g, \ |A(F_{q^i})| = |A'(F_{q^i})|$.
(iv) The Hasse-Weil zeta functions of A and A' coincide.
Although I have not checked in order to answer this question, I think it is likely that proofs -- or references to proofs -- of this fact can be found in at least one of the papers
Waterhouse, William C. Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521--560.
Waterhouse, W. C.; Milne, J. S. Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 53--64. Amer. Math. Soc., Providence, R.I., 1971.