Noise doesn't protect against that, statistics is a thing.
But I think you might overall be right that this requires two colocated processes: the paper talks about how the DMP breaks assumptions made by "the constant-time programming model", and I took this to mean that constant-time algorithms aren't constant-time any more. Reading more closely, I think maybe the issue is that "the constant-time programming model" was also assumed to make secrets safe from cache timing side-channels leaking the secrets to other processes on the same CPU, and this seems like it might be the assumption that's broken by the DMP...
I'll have to read more, I've just skimmed the abstract and introduction so far.