Solution:

Since $(3, m)$ is on the graph of $f(x) = \log_{a} x$, we have:
$$m = \log_{a} 3$$

Similarly, $(x_1, y_1)$ and $(x_2, y_2)$ are on the graph, so:
$$y_1 = \log_{a} x_1 \\ y_2 = \log_{a} x_2$$
Given $y_1 + y_2 = 2m$, so:
$$\log_{a} x_1 + \log_{a} x_2 = 2 \log_{a} 3$$
Using the property $\log_{a} x_1 + \log_{a} x_2 = \log_{a} (x_1 x_2)$:
$$\log_{a} (x_1 x_2) = \log_{a} 3^2 = \log_{a} 9$$
Therefore,
$$x_1 x_2 = 9$$