Distributed Graph Neural Network (GNN) training depends critically on how the underlying graph is partitioned across compute resources. Existing graph partitioners focus either on vertex partitioning or edge partitioning and typically optimize only a single communication objective (edge cut or vertex cut) under a single balance constraint (vertex balance or edge balance). We present SIGMA (Streaming Integrated Graph Partitioning with Multi-objective Awareness), a versatile streaming graph partitioner that supports both vertex and edge partitioning within a unified multi-objective, multi-constraint framework. Depending on the target distributed GNN system, SIGMA can be configured for edgecut-oriented vertex partitioning or vertex-cut-oriented edge partitioning while simultaneously accounting for both vertex and edge balancing. A clustering-based preprocessing stage incorporates global graph structure to improve partition quality while preserving the efficiency and scalability advantages of streaming partitioning. We evaluate SIGMA on six benchmark graphs spanning diverse domains and scales using two distributed GNN training systems: Dist-GNN (edge-partitioned) and DistDGL (vertex-partitioned). Across both settings, SIGMA consistently achieves strong performance, showing its ability to navigate complex trade-offs between partition quality, training efficiency, and memory consumption, frequently outperforming streaming baselines while remaining competitive with high-quality in-memory partitioners such as METIS, KaHIP and HEP. These results demonstrate that a unified streaming partitioner can effectively address the communication, compute, and memory challenges of distributed GNN training across fundamentally different system architectures.