Abstract Nonlocal integral constitutive laws, for application to nano-beams, are investigated in a general setting. Both purely nonlocal and mixture models involving convolutions with averaging kernels are taken into account. Evidence of boundary effects is enlightened by theoretical analysis and numerical computations. Proposed compensation procedures are analyzed, relevant new results are evidenced and confirmed by computations. The strain-driven model and related local-nonlocal mixtures are addressed, with singular phenomena foreseen and numerically quantified. Effectiveness of the recently proposed stress-driven nonlocal elastic model is discussed and illustrated by description of a general solution procedure for nonlocal elastic beams. Comparisons between strain-driven models, stress-driven models and local/nonlocal mixtures are considered from theoretical and computational perspectives. Examples of statically determinate and indeterminate beams are elaborated to show that an effective simulation of scale effects in nano-structures, ensuring existence and uniqueness of solution for any data, is provided by the stress-driven model.