Do two stars on exact opposite sides of a black hole pull on each other gravitationally?

Each star would orbit the barycenter of the black hole and the other star. No information can come from beyond the event horizon, so as far as gravitational information we would only be able to study the system or its components.

Do two stars on exact opposite sides of a black hole pull on each other gravitationally?

I assume you are imagining these stars to be outside the event horizon?

The answer is yes. Let's say it's a black hole with a Schwarzschild radius of 10 km. Then at distances greater than 10 km, it acts exactly like any other object of the same mass. The motion of your entire system is exactly the same as if it was some other object of the same mass. If the distances are large compared to 10 km, then in addition, Newtonian gravity is a good approximation.

If so, how do their gravitational fields go through the black hole?

Physics doesn't work according to Newtonian instantaneous action at a distance. Forces don't reach out along a straight line through space. The gravitational fields in this system are determined by the Einstein field equations, which at distances large compared to 10 km are well approximated by Newtonian gravity.

Can information be sent this way (via gravitational waves)?

Sure. For example, if one of your stars has a planet, then the system will emit gravitational waves (probably very faint ones), and those waves will be detectable at the other star's position. They propagate around the black hole rather than through it. These waves and their propagation are a non-Newtonian effect, and the details of the wave pattern as it arrives at the other star will, I imagine, be different than if you had replaced the black hole with some other type of object.