Consider a wave equation, say in $1+1$ dimensions for $\phi(x,t)$, on a bounded domain, say $x \in (0,L)$ and $t \in \mathbb{R}$, with initial values $\phi(x,0)=u(x)$ and $\partial_t \phi(x,0)=v(x)$ and some boundary conditions.
We know that the solution to the wave equation in the domain of dependence is solely determined by the initial values. Now, if the boundary is outside the domain of dependence the boundary conditions shouldn't affect the solution. But, if we try to solve the wave the equation using the separation of variables (if possible) or using eigenfunctions of the Laplace operator, then the boundary conditions are important because different types of boundary conditions may lead to different eigenfunction, no matter whether the boundary will be inside of the domain of dependence or not.
So what is my confusion here?
Your confusion is that you need first to extend your initial conditions on $(0,L)$ to the whole real line. The way how you extend these conditions depends crucially on the type of boundary conditions at $0$ and $L$.