Let $S^1$ be a circle with a base point $o$. We know that there is an equivalence of categories between: (1) The category of local systems on $S^1$ (2) The category of finite dimensional representations of $\pi_1({S^1,o})$
I wonder if I can have any meaningful statement when the fundamental group is base free, i.e. the cateogry of finite dimensional representations of $\pi_1({S^1})$
You switch between talking about a circle and talking about an arbitrary space and I'm not sure why; just to fix ideas, the equivalence is between the category of local systems of (stuff) on a nice path-connected based space $(X, x)$ and the category of actions of $\pi_1(X, x)$ on (stuff); here (stuff) could be sets, finite sets, vector spaces, finite-dimensional vector spaces, etc.
There is no such thing as a "base-free fundamental group"; we always take the fundamental group with respect to some basepoint. Here the basepoint is needed to define the fiber functor from local systems, given by taking the fiber of the local system at that point. When people say $\pi_1(X)$ without specifying a basepoint, $X$ is path-connected and they mean to take the fundamental group at any basepoint, since the results are all isomorphic; this uniquely specifies an isomorphism class of group but does not do so functorially.
However, we can instead talk about the fundamental groupoid of a not-necessarily-path-connected $X$, $\Pi_1(X)$, which makes sense without a choice of basepoint because we choose all of them. Now the statement is that the category of local systems of (stuff) on a nice space $X$ is equivalent to the category of functors from $\Pi_1(X)$ to the category of (stuff), with the equivalence given by taking the fiber at every point.