One of the best brief descriptions of this process is on one of Wim van Egmond's pages on Molecular Expressions :
During conjugation, two paramecia of compatible mating strains join together along their oral sides and a breakdown of their membranes facilitates the temporary formation of a shared cytoplasmic bridge. The micronuclei of both ciliates undergo several divisions and one from each travels across the bridge, where it fuses with a micronucleus of the opposing party. All other macro- and micronuclei then dissipate, leaving a single nucleus in both cells. Subsequently, the conjugating ciliates pull apart and their zygotic nuclei divide repeatedly to create both types of nuclei. This development is often immediately followed by the creation of new cells via fission.