Maybe this picture is better.
a is the first black hole singularity
A is the region within a's event horizon
b is the second black hole singularity
B is the region within b's event horizon
the masses of the black holes are equal
c is the midpoint between the singularities
If you traveled along the mid-axis (in red), within the event horizons of both black holes,>A: "haha! you are forever trapped within my realm!">B: "not so fast, A, they are forever trapped within MY realm!"
Would these equal gravitational forces keep a person from falling into either singularity a or singularity b, and allow a person to travel the mid-line along its length, into, and out, of the contested overlap region?
Or once they cross into the region in common, they can never come out into "normal" space again?
I know that if you were to fall past the event horizon of a black hole, before you reached the singularity, tidal forces become so severe that you are stretched apart like spaghetti. But what if you were equally between the two black holes at the lagrangian point C? Though the tidal forces would be equal along the mid-line, they would be slightly unequal on either side. But would the small difference still be enough to prevent spaghetti-fication? Or are the tidal forces be so severe (as the distance between A and B decreased), and the imbalance too far apart, that you would be stretched apart anyway?