Keep in mind, in Star Trek, over the years there have been many explanations for Warp-Drive. During TMP the idea proposed by some scientists was similar to Alcubierre's Warp-Drive or perhaps the Bias-Drive; during TNG they altered it to some kind of mass-reducing drive thingie that submerges the ship into the magical world of subspace in which the speed of light is faster and hopping the ship in and out at planck time so the ship is not in warp-space long enough to be affected by the laws of physics in subspace.
The bias-drive or alcubierre type drive would result in an a field in the front of the ship which would compress the fabric of space allowing the ship to traverse large distances without technically going over the speed of light, then stretching it out in the back to normalize the fabric of space. The drive would pull the ship forward at the front and push the ship forward at the rear.
The spatial distortions to allow this to work would be massive -- If you look at the induction-ring the drive is located far out to project a massive space warping field, but the rest of the ship would be fairly inside to avoid extreme tidal forces from the field itself. In Star Trek it would be presumed that the Warp Field would be sufficiently far away from the ship that most of the ship is substantially inside the bubble to avoid this.
A collision of two warp-fields and the interacting tidal-forces and spatial distortions produced by such a drive could be massively destructive.
