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We study the transmission of two discrete memoryless correlated sources, consisting of a common and a private source, over a discrete memoryless multiterminal channel with two transmitters and two receivers. At the transmitter side, the common source is observed by both encoders but the private source can only be accessed by one encoder. At the receiver side, both decoders need to reconstruct the common source, but only one decoder needs to reconstruct the private source. We hence refer to this system by the asymmetric two-user source-channel coding system. We derive a universally achievable lossless joint source-channel coding (JSCC) error exponent pair for the two-user system by using a technique which generalizes Csiszar's type-packing lemma (1980) for the point-to-point (single-user) discrete memoryless source-channel system. We next investigate the largest convergence rate of asymptotic exponential decay of the system (overall) probability of erroneous transmission, i.e., the system JSCC error exponent. We obtain lower and upper bounds for the exponent. As a consequence, we establish a JSCC theorem with single-letter characterization and we show that the separation principle holds for the asymmetric two-user scenario. By introducing common randomization, we also provide a formula for the tandem (separate) source-channel coding error exponent. Numerical examples show that for a large class of systems consisting of two correlated sources and an asymmetric multiple-access channel with additive noise, the JSCC error exponent considerably outperforms the corresponding tandem coding error exponent.