We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C_1\times C_2$, where $C_1$ and $C_2$ are two curves in $\mathbb{A}^n$. First, is there an algorithm to decide this? Second, is it true that if $V$ is of degree $d$ and does contain a product of the form $C_1\times C_2$, then $V$ contains a product of the form $C_1\times C_2$ with $\deg C_1,\deg C_2\leq f(d,n)$, for some function $f$?

Yes, there is such an algorithm. There is an effectively computable constant $N$ such that if $V$ contains a product $S\times T$ where $S,T$ are $N$-point sets, then $V$ contains product of two curves. It is actually true even in the semialgebraic setting. The result is Theorem 1.9 from http://arxiv.org/abs/1207.0705

[ Apologies for answering my own question with a reference to my own paper. When I asked the question, I did not know the answer. ]