Solution:
Answer: $1006$
If there is a chip in every square along a main diagonal, then we need at least $1006$ rows and columns to contain all these chips. We are left to show that $1006$ is sufficient.

Take the $1006$ rows with greatest number of chips. Assume without loss of generality they are the first $1006$ rows. If the remaining $1005$ rows contain at most $1005$ chips, then we can certainly choose $1006$ columns that contain these chips. Otherwise, there exists a row that contains at least $2$ chips, so every row in the first $1006$ rows must contain at least $2$ chips. But this means that there are at least $2 \times 1006 + 1006 = 3018$ chips in total. Contradiction.