Compute memories are memory arrays augmented with dedicated logic to support arithmetic. They support the efficient execution of data-centric computing patterns, such as those characterizing Artificial Intelligence (AI) algorithms. These architectures can provide computing capabilities as part of the memory array structures (In-Memory Computing, IMC) or at their immediate periphery (Near-Memory Computing, NMC). By bringing the processing elements inside (or very close to) storage, compute memories minimize the cost of data access. Moreover, highly parallel (and, hence, high-performance) computations are enabled by exploiting the regular structure of memory arrays. However, the regular layout of memory elements also constrains the data range of inputs and outputs, since the bitwidths of operands and results stored at each address cannot be freely varied. Addressing this challenge, we herein propose a HW/SW co-design methodology combining careful per-layer quantization and inter-layer scaling with lightweight hardware support for overflow-free computation of dot-vector operations. We demonstrate their use to implement the convolutional and fully connected layers of AI models. We embody our strategy in two implementations, based on IMC and NMC, respectively. Experimental results highlight that an area overhead of only 10.5% (for IMC) and 12.9% (for NMC) is required when interfacing with a 2KB subarray. Furthermore, inferences on benchmark CNNs show negligible accuracy degradation due to quantization for equivalent floating-point implementations.