No, they would not counteract eachother's tidal pull because the pull of gravity does not increase linearly with distance. The planet would still have two tides, as water will be attracted to each moon.
The formula for determining gravitational force between two objects includes the gravitational constant, the masses of the two objects, and the square of the distance between the two objects. Because of the formula's inverse proportionality, water on one side of the planet will feel a significantly smaller attraction to a given moon than water on the other side of the planet.
So, if Moon A was on one side of a planet and Moon B was on the opposite side, water on Moon A's side would feel a stronger force towards Moon A than Moon B, leading to a tide on Moon A's side. Similarly, Moon B's side would have its own identical tide, as water on Moon B's side would be more attracted to Moon B than to Moon A.