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It is shown that each conservative non-singular Bernoulli shift is either of type $\mathit{II}_{1}$ or $\mathit{III}_{1}$ . Moreover, in the latter case the corresponding Maharam extension of the shift is a $K$ -automorphism. This extends earlier results obtained by Kosloff for equilibrial shifts. Non-equilibrial shifts of type $\mathit{III}_{1}$ are constructed. We further generalize (partly) the main results to non-singular Markov shifts.

K-property for Maharam extensions of non-singular Bernoulli and Markov shifts