Direct image of the ideal sheaf of the diagonal embedding

Question

Let $X$ be a scheme and $I$ be the ideal sheaf of the diagonal embedding $\Delta:X\hookrightarrow X\times X$. Denote by $p: X\times X\to X$ the projection on the first component. Is it true that $p_*I = 0$? This question boils down to the question what is the pushforward of the quotient map $\mathcal{O}_{X\times X}\to\Delta_*\mathcal{O}_X$ along $p$. It is either $0$ or identity but I am not sure what the correct answer is.