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Multiterminal (MT) source coding refers to separate lossy encoding and joint decoding of multiple correlated sources. Recently, the rate region of both direct and indirect MT source coding in the quadratic Gaussian setup with two encoders was determined. We are thus motivated to design practical MT source codes that can potentially achieve the entire rate region. In this paper, we present two practical MT coding schemes under the framework of Slepian-Wolf coded quantization (SWCQ) for both direct and indirect MT problems. The first, asymmetric SWCQ scheme relies on quantization and Wyner-Ziv coding, and it is implemented via source splitting to achieve any point on the sum-rate bound. In the second, conceptually simpler scheme, symmetric SWCQ, the two quantized sources are compressed using symmetric Slepian-Wolf coding via a channel code partitioning technique that is capable of achieving any point on the Slepian-Wolf sum-rate bound. Our practical designs employ trellis-coded quantization and turbo/low-density parity-check (LDPC) codes for both asymmetric and symmetric Slepian-Wolf coding. Simulation results show a gap of only 0.139-0.194 bit per sample away from the sum-rate bound for both direct and indirect MT coding problems.