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In this paper, we propose distributed algorithms that solve a system of Boolean equations over a network, where each node in the network possesses only one Boolean equation from the system. The Boolean equation assigned at any particular node is a {\em private} equation known to this node only, and the nodes aim to compute the exact set of solutions to the system without exchanging their local equations. We show that each private Boolean equation can be locally lifted to a linear algebraic equation under a basis of Boolean vectors, leading to a network linear equation that is distributedly solvable using existing distributed linear equation algorithms as a subroutine. A number of exact or approximate solutions to the induced linear equation are then computed at each node from different initial values. The solutions to the original Boolean equations are eventually computed locally via a Boolean vector search algorithm. We prove that given solvable Boolean equations, when the initial values of the nodes for the distributed linear equation solving step are i.i.d selected according to a uniform distribution in a high-dimensional cube, our algorithms return the exact solution set of the Boolean equations at each node with high probability. Furthermore, we present an algorithm for distributed verification of the satisfiability of Boolean equations, and prove its correctness. Finally, we show that by utilizing linear equation solvers with differential privacy to replace the in-network computing routines, the overall distributed Boolean equation algorithms can be made differentially private. Under the standard Laplace mechanism, we prove an explicit level of noises that can be injected in the linear equation steps for ensuring a prescribed level of differential privacy.