This paper presents novel convergence results for the Augmented Lagrangian based Alternating Direction Inexact Newton method (ALADIN) in the context of distributed convex optimization. It is shown that ALADIN converges for a large class of convex optimization problems from any starting point to minimizers without needing line-search or other globalization routines. Under additional regularity assumptions, ALADIN can achieve superlinear or even quadratic local convergence rates. The theoretical and practical advantages of ALADIN compared to other distributed convex optimization algorithms such as dual decomposition and the Alternating Direction Method of Multipliers (ADMM) are discussed and illustrated by numerical case studies.