Short answer: yes, you have to use something different, e.g. a Neumann-Neumann method. A good reference is Widlund's book. Non-overlapping methods are based on the principle that, if $u$ solves the Poisson equation in a domain $\Omega$ which has been partitioned into two domains $\Omega_1$, $\Omega_2$, at the shared boundary $\Gamma$ of both domains, $u$ and its gradient have to match up across $\Gamma$.
By contrast, for an overlapping DD method, the intersection of the domains $\Omega_1$ and $\Omega_2$ has a non-empty interior. You can use the interior values of $u$ in $\Omega_1$ as Dirichlet boundary conditions to solve the Poisson equation in $\Omega_2$, then use the interior values in $\Omega_2$ to update the solution in $\Omega_1$, and so forth. If you try and apply an overlapping method to domains that don't actually overlap, there's no room for the solutions on each sub-domain to disagree, and so you'll update neither.
More to the point, the convergence rate of an overlapping DD method goes like $H/h$, where $H$ is the width of the overlap and $h$ is the mesh width*. There's a trade-off: you can have a really small overlap and cheap subdomain solves, but the method won't converge fast; or you can have a large overlap with more expensive subdomain solves, but the whole method converges faster.
*Unless you use a global coarse solve, but now we're getting off topic.