Obviously a group algebra is left-noetherian iff it's right noetherian, let's call it noetherian (as usual).

If $R[G]$ is left noetherian for some nonzero commutative ring $R$ (associative unital), then $G$ is noetherian, i.e. satisfies the max property for subgroups, i.e. every subgroup is finitely generated, or equivalently every ascending sequence of subgroup stabilizes.

Indeed if $H$ is a subgroup, then the kernel of the $R$-module homomorphism $R[G]\to R[G/H]$ is the left ideal $I_H$ consisting of finitely supported sums $\sum\alpha_g\delta_g$ such that $\sum_{g\in g_0H}\alpha_g=0$ for every left coset $g_0H$. Since $R\neq 0$ the map $H\mapsto I_H$ is injective and increasing, whence the result.

Examples of noetherian groups are virtually polycyclic groups, and for them $R[G]$ is noetherian for every noetherian $R$. These are the only known examples with $R[G]$ noetherian (this is a well-known open question).

Still, there exists a few other examples of noetherian groups, first constructed by Olshanskii (Tarski monsters and variants), for which the group algebra is not known to be noetherian.

On the other hand plenty of groups are known not to be noetherian and thus do not have a noetherian group algebra:

- infinitely generated groups
- groups with a non-abelian free subgroup (they contain a free subgroup on countably many generators), e.g. $\mathrm{GL}(n,\mathbf{Z})$ for all $n\ge 2$
- elementary amenable (e.g. solvable) groups that are not virtually polycyclic.
- (thanks to the previous 2 items and Tits' alternative): all linear groups that are not virtually polycyclic

The question about 2-sided noetherianity is a bit more delicate: the obvious obstruction is max-n (maximal condition on normal subgroups). This property fails for many groups (e.g. $\mathrm{GL}(2,\mathbf{Z})$) but holds for many groups (e.g. $\mathrm{GL}(n,\mathbf{Z})$ for $n\ge 3$) and I do not know if their integral group algebra is 2-sided noetherian.