With the ongoing electrification efforts in Europe, Home Energy Management Systems (HEMS) are becoming increasingly relevant as residential energy systems are growing more complex. With the rise of rooftop photovoltaics, home batteries, heat pumps, and electric vehicles with bidirectional charging capabilities, the question of how to coordinate all these components effectively is no longer straightforward. At the same time, how and where HEMS functions are deployed, on local hardware, in the cloud, or across both layers, has significant implications for system reliability, safety, and performance. Current systems can often be classified as some form of hybrid between local and cloud and differences come down to functional allocation. Which HEMS functions should be run locally, which in the cloud, which require a distributed approach and what criteria should drive those decisions? With a mixed-methods approach using semi-structured expert interviews with industry practitioners and simulation studies of different HEMS functionalities, a functional allocation framework was developed. 16 HEMS functions are identified and classified as Offline, Online, or Distributed. The results show that allocation depends primarily on safety-criticality, latency requirements, connectivity dependence, and data requirements. At the same time, each allocation choice involves economic trade-offs that go beyond technical requirements. Distributed and offline configurations require dedicated edge hardware, professional installation, and higher development overhead, while cloud-first architectures offer a faster and cheaper path to a functional product. The framework is therefore not a prescription to maximize local execution, but a tool for making these trade-offs explicit and giving developers and manufacturers a structured basis for understanding what they are giving up in either direction.