Looking for a strategy for finding each other in a crowd, is it better to have one person move and one person stay put, or have both people move?

Suppose we have a $n$ by $n$ grid of squares. Each person begins on a random square. If a person decides to move, then every second, they move 1 square in a random direction, excluding the direction bringing the person to the square they were previously on. If at some time, both people are on the same square, we are done. We want to compare the expected meeting time of either strategy.

This is like a random walk on $\mathbb{Z}^2$, but the boundedness prevents us from taking the difference of their positions and considering that as another random walk. This is also like a random walk on a graph, but the random direction depends not just on the current square, but also the previous one.