As pointed out in the comments, there are many Banach tensor products, but there is indeed at least one which works nicely for $L^p\otimes L^p$.

In general, the algebraic tensor product $X\otimes Y^*$ can be identified with finite rank operators from $Y$ to $X$. When $X=Y=L^2(\mathbb{R})$, taking the completion in the Hilbert-Schmidt norm gives you the space of Hilbert-Schmidt operators on $L^2(\mathbb{R})$, which can be identified with $L^2(\mathbb{R}^2)$.

Similarly, the space of $q$-summing operators from $L^p(\mathbb{R})$ to $L^q(\mathbb{R})$, when $p^{-1} + q^{-1} = 1$, can be identified with $L^p(\mathbb{R}^2)$. (I don't have the reference for this on hand, and don't recall how much it generalizes; I'll check and update later.)

Added later: I don't know if the anonymous poster is still around, but here is the reference.