My comment was a bit wrong, and too brief, so I'm going to expand this into a partial answer.
Suppose you have a set $R$ with operations $+$ and $\times$ such that $(R, +)$ is a group, and $\times$ left-endodistributes over $+$. Let $0$ be the identity under $+$.
Fix $a, b \in R$. Then
$$a \times 0 = a \times (0 + 0) = 0 \times a + a \times 0 \implies 0 \times a = 0.$$
We also have
$$0 = 0 \times (a + 0) = a \times 0 + 0 \times 0 = a \times 0,$$
and
$$a \times b = a \times (b + 0) = b \times a + a \times 0 = b \times a.$$
Therefore, $\times$ is commutative, and therefore distributes over $+$.
The same can be done with left-exodistributivity.
For left-antidistributivity, consider first
$$0 \times 0 = 0 \times (0 + 0) = 0 \times 0 + 0 \times 0 \implies 0 \times 0 = 0.$$
Next, note that
$$0 \times a = 0 \times (a + 0) = a \times 0 + 0 \times 0 = a \times 0.$$
Then,
$$(0 + a) \times (0 + a) = (0 + a) \times 0 + (0 + a) \times a = 0 \times 0 + 0 \times a + a \times 0 + a \times a,$$
which, when combined with the above identity, simplifies to $0 \times a + 0 \times a = 0$. But then,
$$0 = 0 \times a + 0 \times a = a \times (0 + 0) = a \times 0 = 0 \times a.$$
Finally, this again gives us that $\times$ is commutative and distributes over $+$, as
$$a \times (b + 0) = b \times a + 0 \times a = b \times a.$$
While this doesn't mean that left anti/endo/exo-distributivity properties are of no interest, it does mean that, in order to avoid "trivial" examples (i.e. ones where $\times$ distributes), we have to sacrifice a fair amount of structure of the additive magma, which means the result is not going to be as "ring-like" as you might have hoped.
I guess it might also be that the distributive law doesn't have to break down if something else does.
This is one way that can happen:
Let $E$ be the set of closed intervals of $\mathbb R$ containing $0$ (that is, of the form $[a,b]$ with $a,b\in \mathbb R$, and $a\leq 0$ and $b\geq 0$.)
Define $[a,b]\oplus[c,d]=[\min(a,c),\max(b,d)]$ and
$[a,b]\otimes[c,d]=[a+c, b+d]$
It can be checked that this makes $(E, \oplus)$ a commutative monoid with identity $[0,0]$ and $(E,\otimes)$ a monoid with identity also $[0,0]$, and that $\otimes$ distributes over $\oplus$ on both sides.
So $[0,0]$ is indeed invertible with respect to both operations. But, of course, $(E,\oplus)$ is not a group at all: it does not have all additive inverses. This is what permits $[0,0]$ to be nonabsorbing.
This example appears in Section 5.3 of
Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.
I tried to look up pairs of simple operations that don't distribute over each other to see if that could lead me to an example of a structure where you can "divide by zero" in this sense
Those are not hard to find... what if we took $\oplus$ and $\otimes$ to be just regular addition on $\mathbb R$?
Then $\oplus$ does not distribute over $\otimes$, and both make group operations on the set, and they both have the same neutral element, which is invertible under both operations.
Not enforcing distributivity, or something like it to relate the two operations, permits things to break down almost immediately.
Best Answer
Consider linear functions over $\Bbb{R}$ of the form $f(x)=mx+b.$
This is visibly a vector space, and we may go further to obtain an associative algebra-like-object by giving this vector space a "multiplicative" binary operation defined by composition. This defines a left and right binary operation on this vector space.
That is $f_1 =m_1 x + b_1$ and $f_2 =m_2 x + b_2,$ then define
$$f_1\circ f_2 =m_1 m_2 x + m_1b_2+b_1.$$
Further, given $f_3=m_3 x + b_3,$ then
$$f_1\circ f_3 =m_1 m_3 x + m_1b_3+b_1,$$
and $$f_1\circ f_2+f_1\circ f_3 = m_1(m_2 +m_3)x+ m_1(b_2 +b_3)+2b_1.$$
Where $$f_1\circ (f_2+ f_3)= f_1\circ[(m_2+m_3 )x+b_2+b_3] =m_1(m_2+m_3)x+m_1(b_2+b_3)+b_1 $$
so $$[f_1\circ f_2+f_1\circ f_3]-[f_1\circ (f_2+ f_3)]=b_1.$$
Choosing any $b_1\ne0,$ yields an element $f_1$ which will not possess the distributive property.
So we have all the desirable properties of an algebra, without the left or right distributive property.