Large language models (LLMs) power many modern applications, but serving them at scale remains costly and resource-intensive. Current server-centric systems overlook consumer-grade GPUs at the edge. We introduce SpecEdge, an edge-assisted inference framework that splits LLM workloads between edge and server GPUs using a speculative decoding scheme, exchanging only token outputs over the network. SpecEdge employs proactive edge drafting to overlap edge token creation with server verification and pipeline-aware scheduling that interleaves multiple user requests to increase server-side throughput. Experiments show SpecEdge enhances overall cost efficiency by 1.91× through achieving 2.22× server throughput, and reduces inter token latency by 11.24\% compared to a server-only baseline, introducing a scalable, cost-effective paradigm for LLM serving. The code is available at https://github.com/kaist-ina/specedge