Given some monoid $M$ we can form the algebra $\mathbb C[M]$ by considering all formal sums and lineary extending the given multiplication in $M$.
Let $f = \sum_{x\in M} \lambda_x x$ and $m\in M$. Now we have two ways that $M$ can act on $\mathbb C[M]$ on the right, first we can multiply $$ f \cdot m := \sum_{x\in M} \lambda_x xm $$ that is computing in $\mathbb C[M]$, or we can change the coefficients $$ f \star m := \sum_{x\in M} \lambda_{mx} x $$ and both give a right action of $M$ on $\mathbb C[M]$. The first one is essentially the regular action on $\mathbb C[M]$ on itself (i.e. making it into a right module), the second is called the standard representation (each over $\mathbb C$, the names are taken from this paper, page 87 for the term standard representation and page 89 where the (right) regular representation in terms of groups is introduced).
Similar we have two left actions $m \cdot f := \sum_{x\in M} \lambda_x mx$ and $m\star f := \sum_{x\in M} \lambda_{xm} x$.
1) If $M$ is a group, then all those actions give the same representations.
As written on wikipedia if $M$ is group, we have $$ m \star f = \sum_{x\in M} \lambda_{xm} x = \sum_{xm^{-1}\in M} \lambda_{xm^{1-}m} xm^{-1} = \sum_{x\in M} \lambda_x xm^{-1} = f\cdot m^{-1} $$ and similar $f \star m = m^{-1}\cdot f$. If we define $m \circ n := nm^{-1}$ ($m$ acts on $n$) we get a left action on $M$ on itself, and for the mapping $\varphi(x) = x^{-1}$ we have $$ \varphi(mn) = \varphi(n)m^{-1} = m \circ \varphi(n) $$ and as the action $m \cdot f$ is just the linear extension of multiplication from the right this action is equivalent to $m \circ f$ with linear extension. But $m \circ f = f \cdot m^{-1} = m \star f$. Further as the oppositve group is isomorpic to $M$ by $\varphi$ and $m\star f = f\cdot m^{-1}$ from a representational theoretical viewpoint they give the same representations. Hence they all give the same representation theory. $\square$
2) But this does not hold in general.
For example let $M = \{1,a,b\}$ with $b = ab = ba = aa$. Then for every $f \in \mathbb C[M]$ with $f = \lambda_1 1 + \lambda_a a + \lambda_b b$ we have $f\cdot a = \lambda_1 a + (\lambda_a + \lambda_b)b$ and $f\cdot b = (\lambda_1 + \lambda_a+ \lambda_b)b$. Hence a subspace of $\mathbb C[M]$ could just be invariant if it contains the span of $b$, and this span is itself the smallest (irreducible) invariant subspace (which by the way has no complement). On the other side we have \begin{align*} f \star a & = \lambda_{a1}1 + \lambda_{aa}a + \lambda_{ab}b = \lambda_a1 + \lambda_b a + \lambda_b b = \lambda_a 1 + \lambda_b( a + b)\\ f \star b & = \lambda_{b}1 + \lambda_{b}a + \lambda_{b}b = \lambda_b (1 + a + b). \end{align*} hence every invariant subspace contains $1, a+b, 1+a+b$, i.e. the span of $1$ and $a+b$, and this span is the only irreducible invariant subspace.
So in this case both action give quite different representations, they are not equivalent. $\square$
So, what is the relation between those for representations? They all arise quite naturally. Example 1) shows that in the group case they all give the same, but example 2) on the other extreme that they can be different. So when do they all give the same, when does at least two give the same? And when do they all give different representations? And is there anything where this matters? For example when we want to classify all representations for groups, we can use either representation, but in the general case then do we have to look at every representation or is there a single "most universal one" that contains everything?
EDIT: As suggested in the comments, usually left- and right representations are unrelated. So I am still interested in the relations between the two given right representations $f\cdot m$ and $f\star m$, is there any relation in general? (or similar the two left representations).
Suppose that $A$ is an algebra, and $V$ is a right $A$-representation, with action written $(v, a) \mapsto v \cdot a$. Then the dual space $V^* = \mathrm{Hom}(V, \mathbb{C})$ may be naturally equipped with a left $A$-action, written $(a, f) \mapsto a \star f$, defined on a function $f:V\to \mathbb{C}$ by $(a \star f)(v) = f(v \cdot a)$.
Now suppose $A = \mathbb{C}[M]$, and take the right module $V = (A, \cdot)$, where the action is by right multiplication. The dual space $V^*$ may be identified with the vector space of functions $f: M \to \mathbb{C}$, and the star action is $(m \star f)(n) = f(nm)$. It’s not hard to see that this right dot action is the same as the right dot action you defined, and the left star action is the same one as you defined (although you need to take the basis of indicator functions to see the formulas exactly match).
So hopefully this explains what your dots and stars are: the dots are the monoid algebra acting on itself by left or right multiplication, and the stars are the duals of these actions. So asking for a relationship between the left dot and left star actions is kind of like asking for a relationship between the left and right dot actions, which is basically asking for an algebra homomorphism $A \to A^{\text{op}}$. As I mentioned in the comments, there is no good choice for this in general, but there is in special cases ($A$ is a group algebra, $A$ is commutative).
Finally, in your example, $M$ is commutative, and so $A = \mathbb{C}[M]$ is a commutative algebra. Here there is no difference between a left and right action, so both the dot actions are the same, both the star actions are the same, and they are duals of each other. So we can say that they are dual.