Uncertainty and Pareto efficient policies
Pareto efficient allocations can be found by maximizing a weighted average of the utilities of the agents. Let $\lambda$ be the Pareto weight for agent 1 and $1 - \lambda$ the weight for agent 2 (where $\lambda \in [0,1]$). This then gives the following problem:
$$
\max_{x_1, x_2} -\lambda \left[\pi_1(x_1 - b_{11})^2 + \pi_2(x_2 - b_{12})^2\right] - (1-\lambda)\left[ \pi_1 (x_1 - b_{21})^2 + \pi_2 (x_2 - b_{22})^2\right].
$$
The first order conditions with respect to $x_1$ and $x_2$ give:
$$
\lambda \pi_1 2(x_1 - b_{11}) + (1-\lambda) \pi_1 2(x_1 - b_{21}) = 0\\
\lambda \pi_2 2(x_2 - b_{21}) + (1-\lambda) \pi_2 2(x_2 - b_{22}) = 0.
$$
Simplifying gives:
$$
x_1 = \lambda b_{11} + (1-\lambda) b_{21},\\
x_2 = \lambda b_{21} + (1-\lambda) b_{22}.
$$
So $x_1$ and $x_2$ are weighted averages of the bliss points.