This paper deals with the characterization of \(L^p\) Hardy spaces on a domain \(D\) of \({\mathbb R}^d\) by means of Hardy-Stein identities, motivated by the study of martingale Hardy and Bergman spaces, in connection with quadratic variation of martingales and the operator carré du champ. The setting of the paper includes Hardy spaces for the fractional Laplacian \(\Delta^{\alpha/2}\) under conditioning by a fixed \(\alpha\)-harmonic function \(h\), for which the authors derive nonlocal and conditional extensions of the classical Hardy-Stein inequalities.