The buzzword to look for is "maximally almost periodic". This is part of the theory of Bohr Compactifications. Start with a (locally compact) group $G$. I do not believe amenability and (especially) torsion freeness has much if any baring on what following.

The Bohr Compactification is the maximal compact group $K$ for which there is a dense range homomorphism $\theta:G\rightarrow K$. Any finite-dimensional unitary rep of $G$ factors through $\theta$ (by the representation theory of compact groups). It follows that finite-dimensional unitary reps separate the points of $G$ if and only if $\theta$ is injective. By definition, this means that $G$ is "maximally almost periodic" (MAP). One can also construct $K$ using almost periodic functions, and here $G$ is MAP if and only if the almost periodic functions separate the points of $G$.

The Freudenthai-Weil theorem says that for connected locally compact groups, $G$ is MAP if and only if $G=\mathbb R^n \times L$ for some compact $L$. Chapter 16 of Dixmier's book on $C^\ast$-algebras is a good source for all of this.

Searching around MathOverFlow will find examples of groups which are not MAP.