This may sound like a strange question, but say that two stars are relatively next to one another in a binary star system, what would happen to one of the stars if the other went supernova? Would it explode as well, or would it survive the blast?
Remember that although a supernova expels a huge amount of matter, this matter travels outwards in all directions and only a tiny fraction of it impacts on the binary partner. In the case of a core collapse supernova, the vast majority of the energy released is in the form of neutrinos, which will generally fly straight through the binary partner without impact.
In a type 1a supernova, most of the released energy is kinetic. However, as the exploding star is completely disrupted, this "kicks" the other star out of the system (think of swinging a ball on a string in a circle around you, and you suddenly relax your grip on the string), so it's not hanging around for the shock wave. Since this is travelling at 5,000-20,000 km/s, it will eventually (within a few hours) catch up with the escaping star, but will have become substantially attenuated by then.
In addition to Glorfindel's answer, it's worth noting that where the remnant partner of a high-mass X-ray binary is a black hole, it's likely that the progenitor star was also high-mass and collapsed directly to singularity, without a significant explosion. As Felix Mirabel and Irapuan Rodrigues note in Formation of a Black Hole in the Dark: "The observations suggest that high-mass stellar black holes may form promptly, when massive stars disappear silently."
You might also be interested in the answers to these questions on our sibling site [Physics.SE]:
There are certain cases where the remaining star survives; this happened to high-mass X-ray binary star systems, pairs which consist of a neutron star or black hole (the remnant of a supernova) and a massive star. The X-rays are produced when some of the material ejected by the star is captured by the supernova remnant.