Recent efforts in synthetic biology have focussed on the implementation of logical functions within living cells. One aim is to facilitate both internal "re-programming" and external control of cells, with potential applications in a wide range of domains. However, fundamental limitations on the degree to which single cells may be re-engineered have led to a growth of interest in multicellular systems, in which a "computation" is distributed over a number of different cell types, in a manner analogous to modern computer networks. Within this model, individual cell type perform specific sub-tasks, the results of which are then communicated to other cell types for further processing. The manner in which outputs are communicated is therefore of great significance to the overall success of such a scheme. Previous experiments in distributed cellular computation have used global communication schemes, such as quorum sensing (QS), to implement the "wiring" between cell types. While useful, this method lacks specificity, and limits the amount of information that may be transferred at any one time. We propose an alternative scheme, based on specific cell-cell conjugation. This mechanism allows for the direct transfer of genetic information between bacteria, via circular DNA strands known as plasmids. We design a multi-cellular population that is able to compute, in a distributed fashion, a Boolean XOR function. Through this, we describe a general scheme for distributed logic that works by mixing different strains in a single population; this constitutes an important advantage of our novel approach. Importantly, the amount of genetic information exchanged through conjugation is significantly higher than the amount possible through QS-based communication. We provide full computational modelling and simulation results, using deterministic, stochastic and spatially-explicit methods. These simulations explore the behaviour of one possible conjugation-wired cellular computing system under different conditions, and provide baseline information for future laboratory implementations.