What is the stalk at a point of the quotient of a scheme?

Question

This came up when I was talking to @Benjalim on the chat.

Consider the space $X=\operatorname{Spec} \mathbb C[x]$. With the usual structure sheaf, this is a scheme. Let $Y$ be $X$ with the points $(x)$ and $(x-1)$ identified via quotienting (topologically). This is probably not a scheme, but we can still equip it with the pushforward sheaf $f_*\mathcal O_X$, where $f$ is the quotient map $X\rightarrow Y$.

Let the identified point on $Y$, the image of both $(x)$ and $(x-1)$ under the quotient map, be $p$.

Question. What is the stalk at $p$ on $Y$?

We guessed that it was the direct sum of the stalks at $(x)$ and $(x-1)$. We ran into some trouble proving this, however, because one cannot separate $(x)$ and $(x-1)$ with non-intersecting open neighborhoods.

Answer 1

I think it would be the intersection of the two stalks. Any open set around that point has a preimage which contains both previous points, so $x$ and $x-1$ can't be invertible; however, any other element can be inverted, so it should leave the intersection of the stalks.