Some physicists argue that there might not be such thing as independence even with "quasars which are located diametrically opposite to one another"[1].

[1] http://arxiv.org/abs/1112.1811 - page 12

The influence may be negligible, but it's theoretically there. The decision-maker is part of the same system (decision is happening within a system of neurons augmented with electronic tools, all of which exist in the same physical reality).

There are no neurons involved, essentially random number generators are making the decisions and in such experiments usually in the nanosecond range while particles are in flight between source and detector to ensure that the decision can not influence the creation and preparation process. And by using a source of randomness billions of light years away it becomes very unlikely that the particle source and the source of randomness could have interacted in an interesting way because the universe is to young to allow interactions due to the finite speed of light. There is of course still a possibility that an interaction happened when the universe was still young and tiny but you can probably imagine what kind of conspiracy the universe would have to have going on for that to work.

Isn't that just determinism followed to its logical end?