Let \((R, \mathfrak m)\) be a Noetherian local ring and \(M\) a finitely generated \(R\)-module of dimension \(d\). \(M\) is called to be Cohen--Macaulay if \(\text{depth} M=\dim M\) where \(\text{depth} M\) is the length of a maximal \(M\)-sequence in \(\mathfrak m\). Dual to this notion is defined as follows: Let \(L\) be an Artinian \(R\)-module. \(L\) is called to be co-Cohen--Macaulay if \(\text{width} L= \)\(N\)-\(\dim L\) where \(\text{width} L\) is the length of a maximal co-regular sequence in \(\mathfrak m\) and \(N\)-\(\dim L\) is the Noetherian dimension of \(L\). It is well-known that \(H^i_{\mathfrak m}(M)\) is Artinian for all \(i\). Thus it makes sense to study about co-Cohen--Macaulayness of \(H^i_{\mathfrak m}(M)\). In this paper, the authors prove the following: (a) If \(M\) is an unmixed \(R\)-module with \(\text{depth} M \geq d-1\), then \(H^d_{\mathfrak m}(M)\) is co-Cohen--Macaulay of Noetherian dimension \(d\) if and only if \(H^{d-1}_{\mathfrak m}(M)\) is either zero or co-Cohen--Macaulay of Noetherian dimension \(d-2\). (b) If \(\text{width} H^i_{\mathfrak m}(M) \geq i-1\) with \(2\leq i