Nobody in his right mind considers a polynomial ring over anything as a local ring because , well, it just isn't one!
However there are similarities with local rings and indeed there is the following analogue of Nakayama's "lemma":
Graded Nakayama
Let $R=\oplus_{n\in \mathbb N} R_n$ be a positively graded ring and $M=\oplus_{n\in \mathbb Z} M_n$ a graded $R$-module.
Then a (maybe infinite) family $(m_i)$ of homogeneous elements of $M$ generate $M$ as an $R$-module if and only if the residue classes $\bar m_i\in M/R^+M$ generate $ M/R^+M$ as an $R/R^+=R_0$-module.
(As usual $R^+=\oplus_{n \geq 1} R_n$)
[Notice that there is absolutely no finiteness hypothesis on the $R$-module $M$]
The algebraic geometric explanation of the similarity is that a graded ring gives rise to a cone and there is a strong interaction between that cone and the local ring of its vertex.