How to extract noise when the noise on three data sets are near identical?
Given the three measurements $y1$, $y2$ and $y3$ coming from three unknown signals $x1$, $x2$ and $x3$, which we could assume additive to some unknown common noise $x4$, we can write:
$$
y1=x1+x4\\
y2=x2+x4\\
y3=x3+x4
$$
or equivalently:
$$
y=Ax\\
A=[1,0,0,1;0,1,0,1;0,0,1,1]
$$
Again, assuming a enough noisy signal, we can have an standard Recursive Least Squares algorithm for solving the standard LS projection recursively:
$$
\hat x=(A^TA)^{-1}A^Ty
$$
This way, and if and only if the additive noise and the noisy signal suficciencies are met, you will solve for your three signals, and your coupled noise.

Because of the former, you even can express this solution using the Kalman Filter