This letter presents tight upper and lower bounds for minimum number of samples (copies of a quantum state) required to attain a prescribed accuracy (measured by error variance) for scalar parameters estimation using unbiased estimators under quantum local differential privacy for qubits. Particularly, the best-case (optimal) scenario is considered by minimizing the sample complexity over all differentially-private channels; the worst-case channels can be arbitrarily uninformative and render the problem ill-defined. In the small privacy budget <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> regime, i.e., <inline-formula> <tex-math notation="LaTeX">$\epsilon \ll 1$ </tex-math></inline-formula>, the sample complexity scales as <inline-formula> <tex-math notation="LaTeX">$\Theta \text {(}\epsilon ^{-2}$ </tex-math></inline-formula>). This bound matches that of classical parameter estimation under local differential privacy. The lower bound however loosens in the large privacy budget regime, i.e., <inline-formula> <tex-math notation="LaTeX">$\epsilon \gg 1$ </tex-math></inline-formula>. The upper bound for the minimum number of samples is generalized to qudits (with dimension d) resulting in sample complexity of <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}\text {(d}\epsilon ^{-2}$ </tex-math></inline-formula>).