Nanoparticles play a central role across multiple engineering domains, focusing the importance of accurately modeling their mechanical behavior. Given their use in advanced materials and devices, understanding the vibrational response of nanostructures is important. Classical continuum mechanics theories fail to accurately capture the behavior of materials at such scales. Therefore, the development and application of modified or novel theories, considering scale effects, are required. One alternative is non-local elasticity, which includes long-range interactions assuming that the stress at a point is a function of strains at all other points in the domain. Frequency spectra of free standing, traction free sphere- and cube-shaped Si, C, and Ge nanoparticles are investigated using local and non-local elasticity theories. Particle sizes R=2.5, 5.0, 7.0, 10, and 15.0 A, internal material lengths l=1 and 2 A, internal-material-length to particle-size ratios l/R= 0.033, 0.067, 0.10, 0.133, 0.143, 0.20, 0.286, 0.40, 0.50, 0.80, 1.0, and 2.0, and local and non-local weighting factors 0.3-0.7, 0.5-0.5, and 0.8-0.2, were considered. It was found that non-local frequencies are lower than those obtained using local elasticity, indicating a material softening effect introduced by the non-local theory. Additionally, non-local frequencies converge to those calculated using classical local elasticity as the local weighting factor increases and the material-internal-length to particle-size ratio decreases. Besides, the non-local frequency-radius product varies with particle size, indicating that the frequency scale invariance holding in classical elasticity is not valid. Instead, normalized non-local frequencies remain constant for a given material-internal-length to particle-size ratio, regardless of the particle size. This result introduces a novel concept of scale invariance within the framework of non-local elasticity. Finally, modal shapes and frequency degeneracy are identical for both approaches. Classical elasticity provides accurate estimates for the lowest frequencies when the characteristic particle dimension is at least 50 times larger than the material internal material.