This article considers distributed estimation of an unknown deterministic parameter vector in a bandwidth constrained multisensor network with a fusion center (FC). Due to the stringent bandwidth requirements, each sensor compresses its observation as a low-dimensional vector via a linear transformation. Then, the FC linearly combines all received compressed data to estimate the deterministic parameter vector based on the best linear unbiased estimator. The problem of interest is to jointly design the dimension assignment (i.e., the compression dimension of each sensor) and the corresponding compression matrix when the total number of compression dimensions is given. Such a joint design problem is formulated as an optimization problem with rank and linear matrix equality constraints, which is shown to be NP-hard for the first time. In addition, penalty decomposition (PD), successive quadratic upper-bound minimization method of multipliers (SQUM-M), and SQUM-M-block coordinate descent (SQUM-M-BCD) are proposed to solve it approximately. Furthermore, we show that any accumulation point of the sequence generated by the PD satisfies the Karush-Kuhn-Tucker conditions of the equivalent formulation of the joint design problem; the SQUM-M admits the same convergence property as the PD under some conditions. Numerical experiments corroborate the merits of PD and SQUM-M-BCD as compared with existing strategies for the heterogeneous scenario, and further illustrate the effectiveness of the proposed algorithms for the correlated noise case.