This paper is devoted to receiver design in a chaotic direct-sequence spread spectrum (CD3S) digital communication system. The demodulation is achieved through chaos synchronization in an efficient manner thanks to dual unscented Kalman filtering. The problem of carrier phase recovery, frequently neglected in chaos-based communication systems, is addressed either through Costas loop prior to demodulation or by dual estimation of the code, symbol and phase in baseband. The input signal power fluctuations, which often causes large errors in synchronizing chaotic waveforms, is also taken into account. Numerical simulations are provided to show the significance of the proposed receiver for both additive white Gaussian noise (AWGN) channel and a nonstationary channel.