We begin by filling the 10 by 10 grid formed by rows and columns 1 through 10 with any values, which we can do in $\left(2^{10}\right)^{100}=2^{1000}$ ways. Then in column 0, there is at most 1 way to fill in the square in row 10, 2 ways for the square in row 9, down to $2^{10}$ ways in row 0. Similarly, there is 1 way to fill in the square in row 0 and column 10, 2 ways to fill in the square in row 0 and column 9, etc. Overall, the number of ways to fill out the squares in row or column 0 is $2^{1} \cdot 2^{2} \cdot 2^{3} \cdots 2^{9} \cdot 2^{10} \cdot 2^{9} \cdot 2^{8} \cdots 2^{1}=2^{100}$, so the number of possible distinct grids $2^{1000} \cdot 2^{100}=2^{1100}$.