No, as far as I know pasta made with squid ink, often called nero di seppia, should be stable, and the ink doesn't normally bleed into the water.
From my experience after boiling black pasta, even the fresh kind, the water comes out clear like with regular one, with just a bit of clear starch being left behind.
If you boil regular and black pasta mixed together the white pasta should not get significantly tainted with black color, just like when you boil tricolore type of colored pasta (those multi colored green, red and white mixes) the colors don't bleed.
You can (faintly) see in this cooking video that both the boiling water and the rinse water are not darkened.
Black ink also doesn't bleed into sauces after boiling, so you can use light colored sauces without negatively affecting it's appearance, nor does it stain containers like white ceramic of dishes or bowls.
Only situation where you should get bleeding black is for dishes where the squid ink is added to the pasta after boiling, either while stirring in the sauce or along with other ingredients, rather than being mixed into the flour.