Self attention premise is exactly that it isn't context free so it is also incorrect to say that batched requests do not mathematically affect each other. They do, and that's by design.
The GPU can’t do anything with weights while they are in VRAM. They have to be moved into the GPU itself first.
So it is about memory round-trips, but not between RAM and VRAM. It’s the round trips between the VRAM and the registers in the GPU die. When batch processing, the calculations for all batched requests can be done while the model parameters are in the GPU registers. Compared to if they were done sequentially, you would multiply the number of trips between the VRAM and the GPU by the number of individual inferences.
Also, batched prompts and outputs are indeed mathematically independent from each other.
For local hosting, a more likely scenario where you could use batching is if you had a lot of different data you wanted to process (lots of documents or whatever). You could batch them in sets of x and have it complete in 1/x the time.
A less likely scenario is having enough users that you can make the first user wait a few seconds while you wait to see if a second user submits a request. If you do get a second request, then you can batch them and the second user will get their result back much faster than if they had had to wait for the first user’s request to complete first.
Most people doing local hosting on consumer hardware won’t have the extra VRAM for the KV cache for multiple simultaneous inferences though.