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Tunneling two-level systems affect damping, noise, and decoherence in a wide range of devices, including nanoelectromechanical resonators, optomechanical systems, and qubits. Theoretically, this interaction is usually described within the tunneling state model. The dimensions of such devices are often small compared to the relevant phonon wavelengths at low temperatures, and extensions of the theoretical description to reduced dimensions have been proposed, but lack conclusive experimental verification. We have measured the intrinsic damping and the frequency shift in magnetomotively driven aluminum nanoelectromechanical resonators of various sizes at millikelvin temperatures. We find good agreement of the experimental results with a model where the tunneling two-level systems couple to flexural phonons that are restricted to one or two dimensions by geometry of the device. This model can thus be used as an aid when optimizing the geometrical parameters of devices affected by tunneling two-level systems.