We propose a framework to design integrated communication and computing (ICC) receivers capable of simultaneously detecting data symbols and performing over-the-air computing (AirComp) in a manner that: a) is systematically generalizable to any nomographic function, b) scales to a massive number of user equipments (UEs) and edge devices (EDs), c) supports the computation of multiple independent functions (streams), and d) operates in a multi-access fashion whereby each transmitter can choose to transmit either data symbols, computing signals or both. For the sake of illustration, we design the proposed multi-stream and multi-access method under an uplink setting, where multiple single-antenna UEs/EDs simultaneously transmit data and computing signals to a single multiple-antenna base station (BS)/access point (AP). Under the communication functionality, the receiver aims to detect all independent communication symbols while treating the computing streams as aggregate interference which it seeks to mitigate; and conversely, under the computing functionality, to minimize the distortion over the computing streams while minimizing their mutual interference as well as the interference due to data symbols. To that end, the design leverages the Gaussian belief propagation (GaBP) framework relying only on element-wise scalar operations coupled with closed-form combiners purpose-built for the AirComp operation, which allows for its use in massive settings, as demonstrated by simulation results incorporating up to 200 antennas and 300 UEs/EDs. The efficacy of the proposed method under different loading conditions is also evaluated, with the performance of the scheme shown to approach fundamental limiting bounds in the under/fully loaded cases.