Gradient Domain Reconstruction - Scaling Problem
The Poisson equation may suffer from ill-posedness if the boundary condition is not suﬃcient to yield a unique solution to the problem. The most often used boundary condition is Dirichlet boundary condition, in which the values of $I(x,y)$ are specified for those points $(x,y)$ on the boundary of $I$.

In your case, after you stretch your solution to another scale, you got the correct solution. It is possible the solution to your Poisson equation is not unique, and your algorithm just found out the correct one based on some restrained part (such as minimal norm of least square) in the optimization. It is much much more possible that it is already sufficient to determine $I(x,y)$ up to an arbitrary additive constant, and this constant is determined by the solution of Laplacian equation: $$\nabla^{2}\hat I= 0$$