Our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory and focused also by Klement in the Linz2000 closing session. This paper is mainly devoted to solving the functional equations of conditional distributivity between uninorms U_1 and U_2, where both U_1 and U_2 are uninorms with continuous underlying operators. And then, the almost complete characterization of such a pair (U_1,U_2) of aggregation operators is given, excepted that the structure of U_1 on only a small fraction of the unit square. Finally, it is found that there exists the pair (U_1,U_2) satisfying the obtained characterization  but not conditionally distributive under some certain conditions, from another angles, which also shows that the characterization of (U_1,U_2) is only necessary but not sufficient.