Large Language Models (LLMs) are increasingly deployed on converged Cloud and High-Performance Computing (HPC) infrastructure. However, as LLMs handle confidential inputs and are fine-tuned on costly, proprietary datasets, their heightened security requirements slow adoption in privacy-sensitive sectors such as healthcare and finance. We investigate methods to address this gap and propose Trusted Execution Environments (TEEs) as a solution for securing end-to-end LLM inference. We validate their practicality by evaluating these compute-intensive workloads entirely within CPU and GPU TEEs. On the CPU side, we conduct an in-depth study running full Llama2 inference pipelines (7B, 13B, 70B) inside Intel's TDX and SGX, accelerated by Advanced Matrix Extensions (AMX). We derive 12 insights, including that across various data types, batch sizes, and input lengths, CPU TEEs impose under 10% throughput and 20% latency overheads, further reduced by AMX. We run LLM inference on NVIDIA H100 Confidential Compute GPUs, contextualizing our CPU findings and observing throughput penalties of 4-8% that diminish as batch and input sizes grow. By comparing performance, cost, and security trade-offs, we show how CPU TEEs can be more cost-effective or secure than their GPU counterparts. To our knowledge, our work is the first to comprehensively demonstrate the performance and practicality of modern TEEs across both CPUs and GPUs for enabling confidential LLMs (cLLMs).